How can I solve the coupled differential equation with variable coefficients?

Question

Vellapandi M Research Scholar 2020-1-8

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回答： Dinesh Yadav 2020-1-22

I try to solve coupled differential equation in matlab. This is the code I was typed in matlab

syms u(t) v(t)
ode1 = diff(u) == v;
ode2 = diff(v) == ((5/sqrt(2))*exp((1-sqrt(2))*t)-(5/sqrt(2))*exp((1+sqrt(2))*t)+1)*u + (((5*(1-sqrt(2)))/sqrt(2))*exp((1-sqrt(2))*t)-((5*(1+sqrt(2)))/sqrt(2))*exp((1+sqrt(2))*t)+2)*v;
odes = [ode1; ode2]
cond1 = u(0) == 1;
cond2 = v(0) == 0;
conds = [cond1; cond2];
[uSol(t), vSol(t)] = dsolve(odes,conds)

In command window

Warning: Unable to find explicit solution. 
> In dsolve (line 190)
  In coupled (line 8) 
Error using sym/subsindex (line 855)
Invalid indexing or function definition. Indexing must follow MATLAB indexing. Function arguments must be symbolic variables, and function body must be sym expression.
Error in coupled (line 8)
[uSol(t), vSol(t)] = dsolve(odes,conds)
 

Can someone help me with this error? Thank you.

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Vellapandi M Research Scholar 2020-1-9

Thank you once again. I need solution in terms of t. But I get this.

Warning: Unable to find explicit solution.

> In dsolve (line 190)

In coupled (line 8)

uSol =

[ empty sym ]

vSol =

[ ]

David Goodmanson 2020-1-9

that's right. Matlab can't solve it symbolically, but there is always numerically..

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Answer 1

Dinesh Yadav 2020-1-22

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/499393-how-can-i-solve-the-coupled-differential-equation-with-variable-coefficients#answer_411363

在 MATLAB Online 中打开

Hi Vellapandi,

Continuing from David Goodmanson's comment you can solve it numerically. Below is the code for it.

tspan = [0 5];
y0=[1;0];
[uSol, vSol] = ode45(@(t,x) [x(1);((5/sqrt(2))*exp((1-sqrt(2))*t)-(5/sqrt(2))*exp((1+sqrt(2))*t)+1)*x(2) + (((5*(1-sqrt(2)))/sqrt(2))*exp((1-sqrt(2))*t)-((5*(1+sqrt(2)))/sqrt(2))*exp((1+sqrt(2))*t)+2)*x(1)],tspan,y0)

You can change the time limits as per your application.

You can refer to documentation link below to see how to solve system of differential equations.

https://www.mathworks.com/matlabcentral/answers/384186-how-to-solve-two-differential-equations-using-ode45

Hope it helps.