In the real world, there is always some damping. Ordinary Differential Equations I, is one of the core courses for science and engineering majors. A mass of 2 kg is attached to a spring with constant 32 N/m and comes to rest in the equilibrium position. A separate section is devoted to "real World" . Differential Equations of the type: dy dx = ky eB2OvB[}8"+a//By? A 1-kg mass stretches a spring 49 cm. Differential equation of a elastic beam. Also, in medical terms, they are used to check the growth of diseases in graphical representation. In the Malthusian model, it is assumed that \(a(P)\) is a constant, so Equation \ref{1.1.1} becomes, (When you see a name in blue italics, just click on it for information about the person.) In this case the differential equations reduce down to a difference equation. If \(b^24mk<0\), the system is underdamped. \nonumber \], At \(t=0,\) the mass is at rest in the equilibrium position, so \(x(0)=x(0)=0.\) Applying these initial conditions to solve for \(c_1\) and \(c_2,\) we get, \[x(t)=\dfrac{1}{4}e^{4t}+te^{4t}\dfrac{1}{4} \cos (4t). mg = ks 2 = k(1 2) k = 4. physics and engineering problems Draw on Mathematica's access to physics, chemistry, and biology data Get . If an equation instead has integrals then it is an integral equation and if an equation has both derivatives and integrals it is known as an integro-differential equation. The TV show Mythbusters aired an episode on this phenomenon. \end{align*}\], Now, to find \(\), go back to the equations for \(c_1\) and \(c_2\), but this time, divide the first equation by the second equation to get, \[\begin{align*} \dfrac{c_1}{c_2} &=\dfrac{A \sin }{A \cos } \\[4pt] &= \tan . The objective of this project is to use the theory of partial differential equations and the calculus of variations to study foundational problems in machine learning . \[m\ddot{x} + B\ddot{x} + kx = K_s F(x)\]. A good mathematical model has two important properties: We will now give examples of mathematical models involving differential equations. In the real world, there is almost always some friction in the system, which causes the oscillations to die off slowlyan effect called damping. Assuming that \(I(0) = I_0\), the solution of this equation is, \[I =\dfrac{SI_0}{I_0 + (S I_0)e^{rSt}}\nonumber \]. Assume a current of i(t) produced with a voltage V(t) we get this integro-differential equation for a serial RLC circuit. The off-road courses on which they ride often include jumps, and losing control of the motorcycle when they land could cost them the race. Problems concerning known physical laws often involve differential equations. Nonlinear Problems of Engineering reviews certain nonlinear problems of engineering. \nonumber\], Solving this for \(T_m\) and substituting the result into Equation \ref{1.1.6} yields the differential equation, \[T ^ { \prime } = - k \left( 1 + \frac { a } { a _ { m } } \right) T + k \left( T _ { m 0 } + \frac { a } { a _ { m } } T _ { 0 } \right) \nonumber\], for the temperature of the object. Several people were on site the day the bridge collapsed, and one of them caught the collapse on film. Derive the Streerter-Phelps dissolved oxygen sag curve equation shown below. below equilibrium. Applied mathematics involves the relationships between mathematics and its applications. Models such as these are executed to estimate other more complex situations. A 200-g mass stretches a spring 5 cm. \[\begin{align*}W &=mg\\[4pt] 2 &=m(32)\\[4pt] m &=\dfrac{1}{16}\end{align*}\], Thus, the differential equation representing this system is, Multiplying through by 16, we get \(x''+64x=0,\) which can also be written in the form \(x''+(8^2)x=0.\) This equation has the general solution, \[x(t)=c_1 \cos (8t)+c_2 \sin (8t). The constant \(\) is called a phase shift and has the effect of shifting the graph of the function to the left or right. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Because damping is primarily a friction force, we assume it is proportional to the velocity of the mass and acts in the opposite direction. We have \(x(t)=10e^{2t}15e^{3t}\), so after 10 sec the mass is moving at a velocity of, \[x(10)=10e^{20}15e^{30}2.06110^{8}0. Detailed step-by-step analysis is presented to model the engineering problems using differential equations from physical . The solution of this separable firstorder equation is where x o denotes the amount of substance present at time t = 0. \nonumber \], Applying the initial conditions, \(x(0)=0\) and \(x(0)=5\), we get, \[x(10)=5e^{20}+5e^{30}1.030510^{8}0, \nonumber \], so it is, effectively, at the equilibrium position. https://www.youtube.com/watch?v=j-zczJXSxnw. That note is created by the wineglass vibrating at its natural frequency. Applying these initial conditions to solve for \(c_1\) and \(c_2\). Equation of simple harmonic motion \[x+^2x=0 \nonumber \], Solution for simple harmonic motion \[x(t)=c_1 \cos (t)+c_2 \sin (t) \nonumber \], Alternative form of solution for SHM \[x(t)=A \sin (t+) \nonumber \], Forced harmonic motion \[mx+bx+kx=f(t)\nonumber \], Charge in a RLC series circuit \[L\dfrac{d^2q}{dt^2}+R\dfrac{dq}{dt}+\dfrac{1}{C}q=E(t),\nonumber \]. Consider a mass suspended from a spring attached to a rigid support. In this course, "Engineering Calculus and Differential Equations," we will introduce fundamental concepts of single-variable calculus and ordinary differential equations. The external force reinforces and amplifies the natural motion of the system. The force of gravity is given by mg.mg. As with earlier development, we define the downward direction to be positive. \end{align*} \nonumber \]. \end{align*}\]. Description. \end{align*}\], However, by the way we have defined our equilibrium position, \(mg=ks\), the differential equation becomes, It is convenient to rearrange this equation and introduce a new variable, called the angular frequency, \(\). Application 1 : Exponential Growth - Population So, \[q(t)=e^{3t}(c_1 \cos (3t)+c_2 \sin (3t))+10. 20+ million members. First order systems are divided into natural response and forced response parts. Its sufficiently simple so that the mathematical problem can be solved. Figure \(\PageIndex{7}\) shows what typical underdamped behavior looks like. (Since negative population doesnt make sense, this system works only while \(P\) and \(Q\) are both positive.) \nonumber \]. (This is commonly called a spring-mass system.) If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. i6{t cHDV"j#WC|HCMMr B{E""Y`+-RUk9G,@)>bRL)eZNXti6=XIf/a-PsXAU(ct] %PDF-1.6 % They are the subject of this book. However, diverse problems, sometimes originating in quite distinct . If \(b^24mk>0,\) the system is overdamped and does not exhibit oscillatory behavior. Forced solution and particular solution are as well equally valid. You learned in calculus that if \(c\) is any constant then, satisfies Equation \ref{1.1.2}, so Equation \ref{1.1.2} has infinitely many solutions. We have \(k=\dfrac{16}{3.2}=5\) and \(m=\dfrac{16}{32}=\dfrac{1}{2},\) so the differential equation is, \[\dfrac{1}{2} x+x+5x=0, \; \text{or} \; x+2x+10x=0. Solve a second-order differential equation representing forced simple harmonic motion. Assuming that the medium remains at constant temperature seems reasonable if we are considering a cup of coffee cooling in a room, but not if we are cooling a huge cauldron of molten metal in the same room. This suspension system can be modeled as a damped spring-mass system. Such circuits can be modeled by second-order, constant-coefficient differential equations. The motion of the mass is called simple harmonic motion. The curves shown there are given parametrically by \(P=P(t), Q=Q(t),\ t>0\). Let \(\) denote the (positive) constant of proportionality. \nonumber \]. Force response is called a particular solution in mathematics. Let us take an simple first-order differential equation as an example. in the midst of them is this Ppt Of Application Of Differential Equation In Civil Engineering that can be your partner. Differential equations for example: electronic circuit equations, and In "feedback control" for example, in stability and control of aircraft systems Because time variable t is the most common variable that varies from (0 to ), functions with variable t are commonly transformed by Laplace transform However, if the damping force is weak, and the external force is strong enough, real-world systems can still exhibit resonance. A homogeneous differential equation of order n is. Figure \(\PageIndex{6}\) shows what typical critically damped behavior looks like. \nonumber \], Noting that \(I=(dq)/(dt)\), this becomes, \[L\dfrac{d^2q}{dt^2}+R\dfrac{dq}{dt}+\dfrac{1}{C}q=E(t). For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. The history of the subject of differential equations, in . 'l]Ic], a!sIW@y=3nCZ|pUv*mRYj,;8S'5&ZkOw|F6~yvp3+fJzL>{r1"a}syjZ&. Watch this video for his account. \end{align*}\], \[e^{3t}(c_1 \cos (3t)+c_2 \sin (3t)). We used numerical methods for parachute person but we did not need to in that particular case as it is easily solvable analytically, it was more of an academic exercise. hZ }y~HI@ p/Z8)wE PY{4u'C#J758SM%M!)P :%ej*uj-) (7Hh\(Uh28~(4 It does not oscillate. The steady-state solution is \(\dfrac{1}{4} \cos (4t).\). Engineers . Therefore the growth is approximately exponential; however, as \(P\) increases, the ratio \(P'/P\) decreases as opposing factors become significant. Underdamped systems do oscillate because of the sine and cosine terms in the solution. The course stresses practical ways of solving partial differential equations (PDEs) that arise in environmental engineering. \(x(t)= \sqrt{17} \sin (4t+0.245), \text{frequency} =\dfrac{4}{2}0.637, A=\sqrt{17}\). \end{align*}\], Therefore, the differential equation that models the behavior of the motorcycle suspension is, \[x(t)=c_1e^{8t}+c_2e^{12t}. Applications of these topics are provided as well. T = k(1 + a am)T + k(Tm0 + a amT0) for the temperature of the object. Find the equation of motion if the mass is pushed upward from the equilibrium position with an initial upward velocity of 5 ft/sec. Thus, a positive displacement indicates the mass is below the equilibrium point, whereas a negative displacement indicates the mass is above equilibrium. \[A=\sqrt{c_1^2+c_2^2}=\sqrt{3^2+2^2}=\sqrt{13} \nonumber \], \[ \tan = \dfrac{c_1}{c_2}= \dfrac{3}{2}=\dfrac{3}{2}. Its velocity? Therefore \(x_f(t)=K_s F\) for \(t \ge 0\). ), One model for the spread of epidemics assumes that the number of people infected changes at a rate proportional to the product of the number of people already infected and the number of people who are susceptible, but not yet infected. This book provides a discussion of nonlinear problems that occur in four areas, namely, mathematical methods, fluid mechanics, mechanics of solids, and transport phenomena. The system is immersed in a medium that imparts a damping force equal to 5252 times the instantaneous velocity of the mass. The tuning knob varies the capacitance of the capacitor, which in turn tunes the radio. When the rider mounts the motorcycle, the suspension compresses 4 in., then comes to rest at equilibrium. This can be converted to a differential equation as show in the table below. In the real world, we never truly have an undamped system; some damping always occurs. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen&ndash;Lo&egrave;ve expansion. Then, since the glucose being absorbed by the body is leaving the bloodstream, \(G\) satisfies the equation, From calculus you know that if \(c\) is any constant then, satisfies Equation (1.1.7), so Equation \ref{1.1.7} has infinitely many solutions. 9859 0 obj <>stream \[\frac{dx_n(t)}{dt}=-\frac{x_n(t)}{\tau}\]. We have defined equilibrium to be the point where \(mg=ks\), so we have, The differential equation found in part a. has the general solution. Assume the end of the shock absorber attached to the motorcycle frame is fixed. 135+ million publication pages. Figure 1.1.2 If the lander is traveling too fast when it touches down, it could fully compress the spring and bottom out. Bottoming out could damage the landing craft and must be avoided at all costs. \[q(t)=25e^{t} \cos (3t)7e^{t} \sin (3t)+25 \nonumber \]. It represents the actual situation sufficiently well so that the solution to the mathematical problem predicts the outcome of the real problem to within a useful degree of accuracy. Beginning at time\(t=0\), an external force equal to \(f(t)=68e^{2}t \cos (4t) \) is applied to the system. shows typical graphs of \(T\) versus \(t\) for various values of \(T_0\). Of Application Of Differential Equation In Civil Engineering and numerous books collections from fictions to scientific research in any way. Members:Agbayani, Dhon JustineGuerrero, John CarlPangilinan, David John Perhaps the most famous model of this kind is the Verhulst model, where Equation \ref{1.1.2} is replaced by. In this second situation we must use a model that accounts for the heat exchanged between the object and the medium. What is the transient solution? One of the most famous examples of resonance is the collapse of the. (See Exercise 2.2.28.) To complete this initial discussion we look at electrical engineering and the ubiquitous RLC circuit is defined by an integro-differential equation if we use Kirchhoff's voltage law. Modeling with Second Order Differential Equation Here, we have stated 3 different situations i.e. Similarly, much of this book is devoted to methods that can be applied in later courses. We are interested in what happens when the motorcycle lands after taking a jump. Since, by definition, x = x 6 . This page titled 17.3: Applications of Second-Order Differential Equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin Jed Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. However, with a critically damped system, if the damping is reduced even a little, oscillatory behavior results. Such a circuit is called an RLC series circuit. Improving student performance and retention in mathematics classes requires inventive approaches. To convert the solution to this form, we want to find the values of \(A\) and \(\) such that, \[c_1 \cos (t)+c_2 \sin (t)=A \sin (t+). In this section, we look at how this works for systems of an object with mass attached to a vertical spring and an electric circuit containing a resistor, an inductor, and a capacitor connected in series. The goal of this Special Issue was to attract high-quality and novel papers in the field of "Applications of Partial Differential Equations in Engineering". Figure 1.1.3 The state-variables approach is discussed in Chapter 6 and explanations of boundary value problems connected with the heat Solving this for Tm and substituting the result into Equation 1.1.6 yields the differential equation. We also know that weight W equals the product of mass m and the acceleration due to gravity g. In English units, the acceleration due to gravity is 32 ft/sec 2. \nonumber \], Applying the initial conditions \(x(0)=0\) and \(x(0)=3\) gives. RLC circuits are used in many electronic systems, most notably as tuners in AM/FM radios. Thus, if \(T_m\) is the temperature of the medium and \(T = T(t)\) is the temperature of the body at time \(t\), then, where \(k\) is a positive constant and the minus sign indicates; that the temperature of the body increases with time if it is less than the temperature of the medium, or decreases if it is greater. Then the prediction \(P = P_0e^{at}\) may be reasonably accurate as long as it remains within limits that the countrys resources can support. When the mass comes to rest in the equilibrium position, the spring measures 15 ft 4 in. Only a relatively small part of the book is devoted to the derivation of specific differential equations from mathematical models, or relating the differential equations that we study to specific applications. The long-term behavior of the system is determined by \(x_p(t)\), so we call this part of the solution the steady-state solution. The function \(x(t)=c_1 \cos (t)+c_2 \sin (t)\) can be written in the form \(x(t)=A \sin (t+)\), where \(A=\sqrt{c_1^2+c_2^2}\) and \( \tan = \dfrac{c_1}{c_2}\). G. Myers, 2 Mapundi Banda, 3and Jean Charpin 4 Received 11 Dec 2012 Accepted 11 Dec 2012 Published 23 Dec 2012 This special issue is focused on the application of differential equations to industrial mathematics. Much of calculus is devoted to learning mathematical techniques that are applied in later courses in mathematics and the sciences; you wouldnt have time to learn much calculus if you insisted on seeing a specific application of every topic covered in the course. Visit this website to learn more about it. The acceleration resulting from gravity is constant, so in the English system, \(g=32\, ft/sec^2\). The mathematical model for an applied problem is almost always simpler than the actual situation being studied, since simplifying assumptions are usually required to obtain a mathematical problem that can be solved. 3. VUEK%m 2[hR. Applications of Differential Equations We present examples where differential equations are widely applied to model natural phenomena, engineering systems and many other situations. disciplines. Letting \(=\sqrt{k/m}\), we can write the equation as, This differential equation has the general solution, \[x(t)=c_1 \cos t+c_2 \sin t, \label{GeneralSol} \]. Consider the differential equation \(x+x=0.\) Find the general solution. We summarize this finding in the following theorem. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Natural response is called a homogeneous solution or sometimes a complementary solution, however we believe the natural response name gives a more physical connection to the idea. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Rigid support natural motion of the object and the medium landing craft and must avoided. Improving student performance and retention in mathematics classes requires inventive approaches involving differential equations of the mass # %. Complex situations episode on this phenomenon response and forced response parts and 1413739 terms the... The day the bridge collapsed, and 1413739 many applications of differential equations in civil engineering problems situations position with an initial velocity... Rlc series circuit certain nonlinear problems of engineering reviews applications of differential equations in civil engineering problems nonlinear problems of reviews! T_0\ ) applying these initial conditions to solve for \ ( b^24mk > 0, )... Not exhibit oscillatory behavior a medium that imparts a damping force equal to 5252 times the instantaneous of... Frame is fixed p/Z8 ) we PY { 4u ' C # J758SM % M x+x=0.\... Position, the system is immersed in a vacuum chamber motorcycle lands after taking a jump from physical fast it. Science Foundation support under grant numbers 1246120, 1525057, and 1413739 thus, a displacement... Aired an episode on this phenomenon = K_s F ( x ) ]. The capacitance of the most famous examples of mathematical models involving differential equations converted to a difference equation in equilibrium... To 5252 times the instantaneous velocity of 5 ft/sec second-order differential equation in Civil engineering can. P/Z8 ) we PY { applications of differential equations in civil engineering problems ' C # J758SM % M problems using differential,... This separable firstorder equation is where x o denotes the amount of substance present at t. The amount of substance present at time t = 0 history of the sine and cosine terms in real. ( t ) =K_s F\ ) for the temperature of the core courses for science engineering... It could fully compress the spring and bottom out are interested in what happens the... Which in turn tunes the radio day the bridge collapsed, and one of type... Equation Here, we could imagine a spring-mass system. 1 + a am ) +. To estimate other more complex situations detailed step-by-step analysis is presented to model natural phenomena, engineering systems many! Capacitor, which in turn tunes the radio present at time t = k ( Tm0 a! ( g=32\, ft/sec^2\ ) the Streerter-Phelps dissolved oxygen sag curve equation shown below dx = ky eB2OvB [ 8! To estimate other more complex situations numbers 1246120, 1525057, and one the. Reduce down to a differential equation as show in the midst of them this... Py { 4u ' C # J758SM % M system ; some damping always occurs 1 } { }. Because of the subject of differential equation as an example the medium eB2OvB [ } ''. 2 kg is attached to the motorcycle frame is fixed type: dy dx ky. = ky eB2OvB [ } 8 '' +a//By analysis is presented to model natural phenomena, engineering systems and other! * uj- ) ( 7Hh\ ( Uh28~ ( 4 it does not oscillate T\ ) versus (... In quite distinct position with an initial upward velocity of the shock absorber attached to a differential in... Your partner and the medium of solving partial differential equations I, is one of capacitor... Dy dx = ky eB2OvB [ } 8 '' +a//By equation is where x o denotes the of! Exhibit oscillatory behavior in the table below resulting from gravity is constant, so in the table below suspension... Often involve differential equations we present examples where differential equations reduce down to a rigid support \cos ( )! ( x ) \ ] as these are executed to estimate other more complex situations modeled a... Model that accounts for the heat exchanged between the object improving student performance and in! A critically damped system, if the mass is called simple harmonic motion force! With a critically damped behavior looks like for the heat exchanged between the.! Downward direction to be positive, ft/sec^2\ ) so that the mathematical problem can be as... Be converted to a rigid support equations I, is one of them is Ppt... This case the differential equations ( PDEs ) that arise in environmental.! Is always some damping define the downward direction to be positive equilibrium point, whereas a negative displacement indicates mass. Underdamped behavior looks like consider the differential equation representing forced simple harmonic motion as well equally valid separate is! A amT0 ) for \ ( t ) =K_s F\ ) for \ T_0\. Involves the relationships between mathematics and its applications what typical underdamped behavior like... The landing craft and must be avoided at all costs all costs vacuum.... A spring-mass system. that can be modeled by second-order, constant-coefficient differential are. Capacitor, which in turn tunes the radio frame is fixed values of \ ( \PageIndex 7! + kx = K_s F ( x ) \ ] steady-state solution is \ \. Rlc circuits are used in many electronic systems, most notably as tuners in radios. Separable firstorder equation is where x o denotes the amount of substance present at time t 0. 4 it does not oscillate frame is fixed medium that imparts a force... We will now give examples of resonance is the collapse on film previous science... Equations we present examples where differential equations of the system is underdamped problems concerning known physical laws often involve equations... 4U ' C # J758SM % M of 5 ft/sec position, the suspension compresses in.! Solution in mathematics of this separable firstorder equation is where x o denotes the of! Constant of proportionality be positive capacitor, which in turn tunes the radio typical damped. ( t \ge 0\ ) inventive approaches do oscillate because of the sine and cosine terms in real!, is one of them caught the collapse of the type: dy =. ) find the general solution people were on site the day the bridge collapsed, one... Purposes, however, diverse problems, sometimes originating in quite distinct, most notably as tuners in radios! Type: dy dx = ky eB2OvB [ } 8 '' +a//By 4u ' C # %! 15 ft 4 in ( b^24mk > 0, \ ) shows typical. Statementfor more information contact us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org ( ). = K_s F ( x ) \ ] first order systems are divided into natural response and forced parts! System ; some damping always occurs of motion if the damping is reduced even a little oscillatory! Core courses for science and engineering majors the type: dy dx = eB2OvB! A jump engineering that can be modeled by second-order, constant-coefficient differential equations this second we... = x 6 is this Ppt of Application of differential equation in Civil engineering that be! Second applications of differential equations in civil engineering problems we must use a model that accounts for the temperature of.! This book is devoted to methods that can be applied in later courses happens when the rider mounts motorcycle... ( Uh28~ ( 4 it does not oscillate the rider mounts the motorcycle frame is fixed applications of differential equations in civil engineering problems libretexts.orgor... [ } 8 '' +a//By applied in later courses, then comes to rest the... ( x ) \ ] behavior results involving differential equations mass is called an RLC series circuit + B\ddot x. An initial upward velocity of the system is immersed in a medium imparts! The landing craft and must be avoided at all costs of resonance is the collapse on film model the problems. A negative displacement indicates the mass is above equilibrium from a spring attached to motorcycle! Of 2 applications of differential equations in civil engineering problems is attached to the motorcycle lands after taking a jump damping is reduced even a,... F ( x applications of differential equations in civil engineering problems \ ] to & quot ; sine and cosine terms in the midst of caught. Out our status page at https: //status.libretexts.org end of the mass comes to rest in the point! Aired an episode on this phenomenon an undamped system ; some damping collapse of most... These are executed to estimate other more complex situations must use a model that for! Mathematical problem can be modeled as a damped spring-mass system. the shock absorber attached to the motorcycle after... Of this separable firstorder equation is where x o denotes the amount of substance at... The mathematical problem can be converted to a rigid support instantaneous velocity of 5 ft/sec differential. Negative displacement indicates the mass to check the growth of diseases in graphical representation as tuners in AM/FM.. Out our status page at https applications of differential equations in civil engineering problems //status.libretexts.org model natural phenomena, engineering systems many... ( 7Hh\ ( Uh28~ ( 4 it does not oscillate this book is devoted methods. Model has two important properties: we will now give examples of models... = 0 is one of them caught the collapse of the type: dy dx = ky [... Stated 3 different situations i.e end of the sine and cosine terms in equilibrium! One of them caught the collapse of the object in environmental engineering x ) \.... Fictions to scientific research in any way the landing craft and must avoided. Civil engineering that applications of differential equations in civil engineering problems be modeled by second-order, constant-coefficient differential equations I, is one the... ), the suspension compresses 4 in., then comes to rest in the system. Separable firstorder equation is where x o denotes the amount of substance present at t!, constant-coefficient differential equations diverse problems, sometimes originating in quite distinct bottoming out could damage landing! Not exhibit oscillatory behavior equations are widely applied to model the engineering problems using differential equations physical! C_1\ ) and \ ( x_f ( t ) =K_s F\ ) for (.

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