Plotting solutions of differential equations in Simulink

The following Simulink model solves the differential equation t - 4x(t) = x''(t), and generates the plot below for x(t). But the plot doesn't look like a plot of t/4 + C1 * cos(2t) -C2*sin(2t), which is x(t). For example, if C1 and C2 are set equal to 1 a different plot is generated. How is the Simulink plot to be interpreted? No initial conditions are specified and it appears that x(0) = 0 and x'(0) = 0 are assumed but it seems that with these initial conditions the differential equation has no solution because the output of Matlab when I specified these initial conditions, or only the condition x(0) == 0, to solve the differential equation with dsolve was empty.

Answer 1

[Tushar Behera]

Hi Aleem Andrew,

I believe you are trying to implement t-4x(t)=x’’(t) in Simulink.

Your model seems to be correct, and the Simulink engine is trying to solve the equation with default initial conditions as x(0)=0 and x’(0)=0. The output for x(t) for the above differential equation seems to be as below,

This is the curve for the equation x(t)=t/4-sin(2*t)/8. This can be further verified by using “dsolve” for solving the above said equation.

syms x(t) a b 
eqn = diff(x,t,2) == t-4*x;
Dx = diff(x,t); 
cond = [x(0)==0, Dx(0)==0];% initial condition x(0)=0 and x’(0)=0
xSol(t) = dsolve(eqn,cond);
t=1:0.2:10; 
plot(xSol(t))

The solution x(t)= t/4 + C1*cos(2*t) - C2*sin(2*t) is when initial conditions are not present for the differential equation. That’s why the plot seems different as it’s for x(t)=t/4-sin(2*t)/8. I hope this helps resolve your query.

Here is the link for additional documentation:

Solve system of differential equations - MATLAB dsolve (mathworks.com)

Thanks and regards,

Tushar Behera