Differential Equations

Math Review with Matlab: Differential Equations 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

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 • Using the ODE45 Solver

Solving a 1st Order DE • The Matlab command used to solve differential equations is dsolve • Consider the differential equation: • The general solution is given by: • Verify the solution using dsolve command

Solving a DifferentialEquation in Matlab » syms y t » ys=dsolve('Dy+2*y=12') ys = 6+exp(-2*t)*C1 • C1 is a constant which is specified by way of the initial condition • Dy means dy/dt and D2y means d2y/dt2 etc

Verify Results • Verify results given y(0) = 9 » ys=dsolve('Dy+2*y=12','y(0)=9') ys = 6+3*exp(-2*t)

Solving a 2nd Order DE » syms c y » ys=dsolve('D2y=-c^2*y') ys = C1*sin(c*t)+C2*cos(c*t) • Find the general solution of:

Solving SimultaneousDifferential Equations Example • Solve the following set of differential equations: • Syntax for solving simultaneous differential equations is: dsolve('equ1', 'equ2',…)

General Solution • Given the equations: • The general solution is given by:

Matlab Verification • Given the equations: • General solution is: » 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)

Initial Conditions • Solve the previous system with the 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)

Non-Linear Differential Equation Example • Solve the differential equation: • 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))

Specifying the Independent Parameter of a Differential Equation • If another independent variable, other than t, is used, it must be introduced in the dsolve command • Solve the differential equation: » y=dsolve('Dy+2*y=12','x') y = 6+exp(-2*x)*C1

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: • No closed form solution exists • Use the ode45 command to get a numerical solution

Rewrite Differential Equation • Rewrite in the following form

Create a New Function • Create a Matlab function evalxdot to evaluate and numerically in terms of x1 and x2. function xdot=evalxdot(t,x) %x=[x1, x2] %xdot=[dx1/dt, dx2/dt]; xdot=[x(2); -9*sin(x(1))];

ODE45 • 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

Returning t, y and dy/dt • 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')

Plot of Solution

Omit Output Arguments • We can run the solver again without output arguments • 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')

Plot of Results

Summary • The symbolic toolbox can be used to find the closed form solutions for differential equations where they exist • The symbolic toolbox can be simultaneously solve a system of differential equations • Other Matlab commands can be used to numerically solve systems of differential equations if closed forms do not exist

Differential Equations