Asked by Tejas Adsul
on 15 May 2018 at 14:39

I have the following system of differential equations, and I am not able to understand the best way to go about them. I tried using a couple of functions like dsolve, ode45, etc., but most of them give errors that I am not able to understand.

x(1)=0.3; x(3)=0.9; y(1)=0.4; y(3)=0.8; A=10; p=10; syms X1 X2; num = ((x(3)-X1)-(X1-x(1)))*X1 + ((y(3)-X2)-(X2-y(1)))*X2; den = sqrt((x(3)-X1)-(X1-x(1))^2 + (y(3)-X2)-(X2-y(1))^2); Em = -A*(X1-x(1))*(x(3)-X1) - A*(X2-y(1))*(y(3)-X2) + p*num/den; dXdt = [-diff(Em,X1); -diff(Em,X2)];

I would like to get X1 and X2 as a function of time. If I define syms X1(t) and X2(t), I get the error 'All arguments, except for the first one, must not be symbolic functions.' The above code, when run, gives me expressions in terms of X1 and X2. I need solutions of X1 and X2 in terms of t, where dX1dt = -diff(Em,X1), dX2dt = -diff(Em,X2).

Any help is appreciated. Thank you!

Answer by Stephan Jung
on 15 May 2018 at 20:59

Edited by Stephan Jung
on 15 May 2018 at 21:04

Accepted Answer

Hi,

does this work for your purpose?

x1=0.3; x3=0.9; y1=0.4; y3=0.8; A=10; p=10;

syms X_1 X_2 X1(t) X2(t); num = ((x3-X_1)-(X_1-x1))*X_1 + ((y3-X_2)-(X_2-y1))*X_2; den = sqrt((x3-X_1)-(X_1-x1)^2 + (y3-X_2)-(X_2-y1)^2); Em = -A*(X_1-x1)*(x3-X_1) - A*(X_2-y1)*(y3-X_2) + p*num/den;

dEm_dX_1 = diff(Em,X_1); dEm_dX_2 = diff(Em,X_2);

dEm_dX_1 = (subs(dEm_dX_1, [X_1 X_2], [X1 X2])); dEm_dX_2 = (subs(dEm_dX_2, [X_1 X_2], [X1 X2]));

ode1 = diff(X1,t) == -dEm_dX_1; ode2 = diff(X2,t) == -dEm_dX_2;

ode = matlabFunction([ode1; ode2])

ode is a function handle:

ode =

function_handle with value:

@(t)[diff(X1(t),t)==X1(t).*-2.0e1+(X1(t).*4.0e1-1.2e1).*1.0./sqrt(-X1(t)-X2(t)-(X2(t)-2.0./5.0).^2-(X1(t)-3.0./1.0e1).^2+1.7e1./1.0e1)+(X1(t).*2.0+2.0./5.0).*(X1(t).*(X1(t).*2.0-6.0./5.0).*1.0e1+X2(t).*(X2(t).*2.0-6.0./5.0).*1.0e1).*1.0./(-X1(t)-X2(t)-(X2(t)-2.0./5.0).^2-(X1(t)-3.0./1.0e1).^2+1.7e1./1.0e1).^(3.0./2.0).*(1.0./2.0)+1.2e1;diff(X2(t),t)==X2(t).*-2.0e1+(X2(t).*4.0e1-1.2e1).*1.0./sqrt(-X1(t)-X2(t)-(X2(t)-2.0./5.0).^2-(X1(t)-3.0./1.0e1).^2+1.7e1./1.0e1)+(X2(t).*2.0+1.0./5.0).*(X1(t).*(X1(t).*2.0-6.0./5.0).*1.0e1+X2(t).*(X2(t).*2.0-6.0./5.0).*1.0e1).*1.0./(-X1(t)-X2(t)-(X2(t)-2.0./5.0).^2-(X1(t)-3.0./1.0e1).^2+1.7e1./1.0e1).^(3.0./2.0).*(1.0./2.0)+1.2e1]

depending on time, which should be able to solve like you wanted to do.

Running it as a live script gives:

That's what you wanted to achieve?

Best regards

Stephan

Tejas Adsul
on 15 May 2018 at 22:44

Thank you very much! It helped a lot.

Stephan Jung
on 15 May 2018 at 22:54

Please note that i have assumed initial conditions:

X1(0)=0

X2(0)=0

You have to check this...i dont have an idea if this is correct.

Best regards

Stephan

Tejas Adsul
on 16 May 2018 at 18:06

By the way, how did you write down the expression of ode(1) and ode(2) in the function odefun?

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Answer by Stephan Jung
on 15 May 2018 at 20:16

Edited by Stephan Jung
on 15 May 2018 at 20:18

Hi,

did i understand right:

x1=0.3; x3=0.9; y1=0.4; y3=0.8; A=10; p=10;

syms X1 X2; num = ((x3-X1)-(X1-x1))*X1 + ((y3-X2)-(X2-y1))*X2; den = sqrt((x3-X1)-(X1-x1)^2 + (y3-X2)-(X2-y1)^2); Em = -A*(X1-x1)*(x3-X1) - A*(X2-y1)*(y3-X2) + p*num/den;

dEm_dX1 = diff(Em,X1) dEm_dX2 = diff(Em,X2)

dX1dt = -dEm_dX1 dX2dt = -dEm_dX2

This is what you wanted to do?

gives:

dX1dt =

(40*X1 - 12)/(17/10 - X2 - (X2 - 2/5)^2 - (X1 - 3/10)^2 - X1)^(1/2) - 20*X1 + ((2*X1 + 2/5)*(10*X1*(2*X1 - 6/5) + 10*X2*(2*X2 - 6/5)))/(2*(17/10 - X2 - (X2 - 2/5)^2 - (X1 - 3/10)^2 - X1)^(3/2)) + 12

dX2dt =

(40*X2 - 12)/(17/10 - X2 - (X2 - 2/5)^2 - (X1 - 3/10)^2 - X1)^(1/2) - 20*X2 + ((2*X2 + 1/5)*(10*X1*(2*X1 - 6/5) + 10*X2*(2*X2 - 6/5)))/(2*(17/10 - X2 - (X2 - 2/5)^2 - (X1 - 3/10)^2 - X1)^(3/2)) + 12

Can this be correct?

Best regards

Stephan

Tejas Adsul
on 15 May 2018 at 20:23

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## 2 Comments

## Stephan Jung (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/400867-help-solving-a-system-of-differential-equations#comment_568307

Hi,

is it correct, that x and y are depending on t --> so: x(t) and y(t)?

Best regards

Stephan

## Tejas Adsul (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/400867-help-solving-a-system-of-differential-equations#comment_568308

The x and y whose values have been mentioned are constants. The variables X1 and X2 do depend on time. However, what I can do is ignore this time dependence, find the diff(Em,X1) and diff(Em,X2) expressions, then consider the time dependence by writing out the ode diff(X1,t) = -diff(Em,X1) and diff(X2,t) = -diff(Em,X2). I was able to do the first part, and am stuck with the second part.

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