1.Differential equation & LAPLACE TRANSFORmation with MATLAB RAVI JINDAL Joint Masters, SEGE (M1) Second semester B.K. Birla institute of Engineering & Technology, Pilani

2. Differential Equations with MATLAB MATLAB has some powerful features for solving differential equations of all types. We will explore some of these features for the Constant Coefficient Linear Ordinary Differential Equation forms. The approach here will be that of the Symbolic Math Toolbox. The result will be the form of the function and it may be readily plotted with MATLAB. 3. Symbolic Differential Equation Terms 2 2 n n y dy dt d y dt d y dt y Dy D2y Dny 4. Finding Solutions to Differential Equations Solving a First Order Differential Equation Solving a Second Order Differential Equation Solving Simultaneous Differential Equations Solving Nonlinear Differential Equations Numerical Solution of a Differential Equation 5. Solving a 1st Order DE Consider the differential equation: 122 =+ y dt dy The general solution is given by: The Matlab command used to solve differential equations is dsolve . Verify the solution using dsolve command 6. Solving a Differential Equation in Matlab C1 is a constant which is specified by way of the initial condition Dy means dy/dt and D2y means d2y/dt2 etc syms y t ys=dsolve('Dy+2*y=12') ys =6+exp(-2*t)*C1 7. Verify Results Verify results given y(0) = 9 ys=dsolve('Dy+2*y=12','y(0)=9') ys = 6+3*exp(-2*t) 39)0( 1 == Cy 8. Solving a 2nd Order DE 8 Find the general solution of: 02 2 2 =+ yc dt yd )cos()sin()( 21 ctCctCty += syms c y ys=dsolve('D2y = - c^2*y') ys = C1*sin(c*t)+C2*cos (c*t) 9. 9 Solve the following set of differential equations: Solving Simultaneous Differential Equations Example yx dt dx 43 += yx dt dy 34 += Syntax for solving simultaneous differential equations is: dsolve('equ1', 'equ2',) 10. The general solution is given by: General Solution )4sin()4cos()( 3 2 3 1 tectectx tt += )4cos()4sin()( 3 2 3 1 tectecty tt += yx dt dx 43 += yx dt dy 34 += Given the equations: 11. Matlab Verification syms x y t [x,y]=dsolve('Dx=3*x+4*y','Dy=-4*x+3*y') x = exp(3*t)*(cos(4*t)*C1+sin(4*t)*C2) y = -exp(3*t)*(sin(4*t)*C1-cos(4*t)*C2) yx dt dx 43 += yx dt dy 34 += Given the equations: General solution is: )4sin()4cos()( 3 2 3 1 tectectx tt += )4cos()4sin()( 3 2 3 1 tectecty tt += 12. Solve the previous system with the initial conditions: Initial Conditions 0)0( =x 1)0( =y [x,y]=dsolve('Dx=3*x+4*y','Dy=-4*x+3*y', 'y(0)=1','x(0)=0') x = exp(3*t)*sin(4*t) y = exp(3*t)*cos(4*t) )4cos( )4sin( 3 3 tey tex t t = = 13. Non-Linear Differential Equation Example Solve the differential equation: 2 4 y dt dy = Subject to initial condition: 1)0( =y syms y t y=dsolve('Dy=4-y^2','y(0)=1') y=simplify(y) y = 2*(3*exp(4*t)-1)/(1+3*exp(4*t)) ( ) t t e e ty 4 4 31 132 )( + = 14. If another independent variable, other than t, is used, it must be introduced in the dsolve command Specifying the Independent Parameter of a Differential Equation 122 =+ y dx dy y=dsolve('Dy+2*y=12','x') y = 6+exp(-2*x)*C1 Solve the differential equation: x eCxy 2 16)( += 15. Numerical Solution Example Not all non-linear differential equations have a closed form solution, but a numerical solution can be found Solve the differential equation: Subject to initial conditions: 0)sin(92 2 =+ y dt yd 1)0( =y 0)0( = y 16. 16 Rewrite Differential Equation yx =1 == 12 xyx )sin(9 )sin(9 12 2 xx yyx = == 0)sin(92 2 =+ y dt yd 1)0()0(1 == yx 0)0()0(2 == yx Rewrite in the following form )sin(92 2 yy dt yd == 17. 17 Solve DE with MATLAB. >> y = dsolve ('D2y + 3*Dy + 2*y = 24', 'y(0)=10', 'Dy(0)=0') y = 12+2*exp(-2*t)-4*exp(-t) >> ezplot(y, [0 6]) 2 2 3 2 24 d y dy y dt dt + + = (0) 10y = '(0) 0y = 18. Definition of Laplace Transformation: Let f(t) be a given function defined for all t 0 , then the Laplace Transformation of f(t) is defined as Here, L = Laplace Transform Operator. f(t) =determining function, depends on t . F(s)= Generating function, depends on s . 19. Differential equations Input excitation e(t) Output response r(t) Time Domain Frequency Domain Algebraic equations Input excitation E(s) Output response R(s) Laplace Transform Inverse Laplace Transform The Laplace Transformation 20. Laplace Transforms with MATLAB Calculating the Laplace F(s) transform of a function f(t) is quite simple in Matlab . First you need to specify that the variable t and s are symbolic ones. This is done with the command >> syms t s The actual command to calculate the transform is >> F = Laplace (f , t , s) 21. example for the function f(t) >> syms t s >> f=-1.25+3.5*t*exp(-2*t)+1.25*exp(-2*t); >> F = laplace ( f , t , s) F = -5/4/s+7/2/(s+2)^2+5/4/(s+2) >> simplify(F) ans = (s-5)/s/(s+2)^2 >> pretty (ans) 22. Inverse Laplace Transform The command one uses now is ilaplace . >> syms t s >> F=(s-5)/(s*(s+2)^2); >> ilaplace(F) ans = -5/4+(7/2*t+5/4)*exp(-2*t) >> simplify(ans) ans = -5/4+7/2*t*exp(-2*t)+5/4*exp(-2*t) >> pretty(ans) - 5/4 + 7/2 t exp(-2 t) + 5/4 exp(-2 t) 23. Reference http://www.mathworks.in/help/symbolic/simpli https://www.google.co.in/#q=laplace+transform+ 24. Thank You