How to Solve Second Order ODE with 2 dependent variables

Question

Yanuar Rizki Pahlevi 2021-1-14

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/717103-how-to-solve-second-order-ode-with-2-dependent-variables

回答： Ariadna 2022-5-5

Hi guys, I am trying to solve this second order ODE, I followed this step

https://www.mathworks.com/help/symbolic/solve-differential-equation-numerically-1.html

diff(x,2)== diff(x)-y
Vx=odeToVectorField(diff(x,2)== diff(x)-y)
[Y[2]; Y[2] - y(t)] %result from odeToVectorField
 
diff(y,2)== diff(y)-x
Vy=odeToVectorField(diff(x,2)== diff(x)-y)
[[Y[2]; Y[2] - x(t)]] %result from odeToVectorField

i tried using odeToVectorField to make it in first order and got 2 vectors. but then I dont understand how to make this to work since on the vector from first DE, there is variable y(t) which always updated during calculation.. it also happened for vector from second DE..

I tried using this step

https://www.mathworks.com/help/symbolic/solve-a-system-of-differential-equations.html

by creating

ode1=diff(x1)==x2;
ode2=diff(x2)==diff(x1)-y1
ode3=diff(y1)==y2
ode4=diff(y2)==diff(y1)-x1
odes1=[ode1;ode2]
odes2=[ode3;ode4]

but dsolve did not work for this.

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Answer 1

Divija Aleti 2021-1-20

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https://ww2.mathworks.cn/matlabcentral/answers/717103-how-to-solve-second-order-ode-with-2-dependent-variables#answer_602029

在 MATLAB Online 中打开

Hi Yanuar,

Two second order ODE's can directly be solved by using 'dsolve'. Have a look at the following code:

syms x(t) y(t)
ode1 = diff(x,2)==diff(x)-y;
ode2 = diff(y,2)==diff(y)-x;
odes = [ode1;ode2];
S = dsolve(odes)
S.x
S.y

Another method would be to convert the two second order ODEs into four first order ODEs and then solve using dsolve.

Let x1 = x, y1 = y

dx1/dt = x2, dy1/dt = y2

dx2/dt = x2 - y1, dy2/dt = y2 - x1

syms x1(t) x2(t) y1(t) y2(t)
ode1=diff(x1)==x2;
ode2=diff(x2)==x2-y1;
ode3=diff(y1)==y2;
ode4=diff(y2)==y2-x1;
odes=[ode1;ode2;ode3;ode4];
S = dsolve(odes);
S.x1 % As x1 = x
S.y1 % As y1 = y