2. Differential Equations Math Review with Matlab: Finding Solutions to Differential Equations S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn

3. Differential Equations:Finding Solutions to DEs 2 Finding Solutions to Differential Equations n Solving a First Order Differential Equation Solving a First Order Differential Equation n Solving a Second Order Differential Equation Solving a Second Order Differential Equation n Solving Simultaneous Differential Equations Solving Simultaneous Differential Equations n Solving Nonlinear Differential Equations Solving Nonlinear Differential Equations n Numerical Solution of a Differential Equation Numerical Solution of a Differential Equation n Using the ODE45 Solver Using the ODE45 Solver

4. Differential Equations:Finding Solutions to DEs 3 n Consider the differential equation: Solving a 1 st Order DE n The general solution is given by: The Matlab command used to solve differential equations is dsolve Verify the solution using dsolve command

5. Differential Equations:Finding Solutions to DEs 4 Solving a Differential Equation in Matlab n C1 is a constant which is specified by way of the initial condition n Dy means dy/dt and D 2 y means d 2 y/dt 2 etc » syms y t » ys=dsolve('Dy+2*y=12') ys = 6+exp(-2*t)*C1

6. Differential Equations:Finding Solutions to DEs 5 n Verify results given y(0) = 9 Verify Results » ys=dsolve('Dy+2*y=12','y(0)=9') ys = 6+3*exp(-2*t)

7. Differential Equations:Finding Solutions to DEs 6 n Find the general solution of: Solving a 2 nd Order DE » syms c y » ys=dsolve('D2y=-c^2*y') ys = C1*sin(c*t)+C2*cos(c*t)

8. Differential Equations:Finding Solutions to DEs 7 n Solve the following set of differential equations: Solving Simultaneous Differential Equations Example n Syntax for solving simultaneous differential equations is: dsolve( ' equ1 ', ' equ2 ',…)

9. Differential Equations:Finding Solutions to DEs 8 n The general solution is given by: General Solution n Given the equations:

10. Differential Equations:Finding Solutions to DEs 9 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) n Given the equations: n General solution is:

11. Differential Equations:Finding Solutions to DEs 10 n Solve the previous system with the initial conditions: Initial Conditions » [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)

12. Differential Equations:Finding Solutions to DEs 11 Non-Linear Differential Equation Example n Solve the differential equation: n Subject to initial condition: » 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))

13. Differential Equations:Finding Solutions to DEs 12 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 » y=dsolve('Dy+2*y=12','x') y = 6+exp(-2*x)*C1 n Solve the differential equation:

14. Differential Equations:Finding Solutions to DEs 13 Numerical Solution Example n Not all non-linear differential equations have a closed form solution, but a numerical solution can be found n No closed form solution exists Use the ode45 command to get a numerical solution n Solve the differential equation: n Subject to initial conditions:

15. Differential Equations:Finding Solutions to DEs 14 Rewrite Differential Equation n Rewrite in the following form

16. Differential Equations:Finding Solutions to DEs 15 Create a Matlab function evalxdot to evaluate Create a New Function andnumerically in terms of x 1 and x 2. function xdot=evalxdot(t,x) %x=[x1, x2] %xdot=[dx1/dt, dx2/dt]; xdot=[x(2); -9*sin(x(1))];

17. Differential Equations:Finding Solutions to DEs 16 ODE45 n ODE45 is used to solve non-stiff differential equations [T,Y] = ODE45('F',TSPAN,Y0 ) T = Time vector Y = Output corresponding to time vector F = Function name TSPAN = Simulation duration Y0 = Initial conditions If the left hand side [T,Y] of the output arguments is omitted, Matlab solves it and plots it

18. Differential Equations:Finding Solutions to DEs 17 Returning t, y and dy/dt n Run the solver with the input and output arguments specified » [t,y]=ode45('evalxdot',10,[1 0]); » plot(t,y) » xlabel('Time (sec)'); » ylabel('Amplitude'); » title('Numerical Solution'); » legend('Y','dY/dt')

19. Differential Equations:Finding Solutions to DEs 18 Plot of Solution

20. Differential Equations:Finding Solutions to DEs 19 Omit Output Arguments n We can run the solver again without output arguments n Omitting the output arguments causes Matlab to plot the results » ode45('evalxdot',10,[1 0]); » xlabel('Time (sec)'); » ylabel('Amplitude'); » title('Numerical Solution'); » legend('Y','dY/dt')

21. Differential Equations:Finding Solutions to DEs 20 Plot of Results

22. Differential Equations:Finding Solutions to DEs 21 Summary n The symbolic toolbox can be used to find the closed form solutions for differential equations where they exist n The symbolic toolbox can be simultaneously solve a system of differential equations n Other Matlab commands can be used to numerically solve systems of differential equations if closed forms do not exist