solve a partial differential equation

Hi！ I want to solve a second partial differential equation.

The general solution is

Which command should I use to get the general solution? I think dsolve doesn't work well. Thanks for helping me.

[madhan ravi]

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Answer 1

[Stephan]

Hi,

the following code solves the equation with the initial conditions:

y(0)=0
dy/dx(0)=0

Then it rewrites the result in terms of sin ans cos:

syms y(x) x E I N P c l Dyt(x)
eqn = E*I*diff(y,x,2)+N*y == -P*c/l*x;
[eqns, vars] = reduceDifferentialOrder(eqn,y);
conds = [y(0)==0,Dyt(0)==0];
sol = dsolve(eqns,conds);
sol_sin = simplify(rewrite(sol.y,'sin'))

I think that was the question.

Best regards

Stephan

[Yuchi Kang]

Thanks for helping me. But it seems wrong.

In addition, reduceDifferentialOrder is used here. Could you please tell me a little bit about the reason?

Best regards

Yuchi Kang

[Stephan]

For me the code runs fine. Is there a copy paste mistake? Which Release do you use?

[Yuchi Kang]

Thanks for your reply. I have checked carefully, there is no paste mistake. R2017a is used.

[madhan ravi]

Works for me

[Stephan]

Im using R2018b - let me check this. Maybe there is a difference in dsolve.

Can you show us the result of typing this in commmand line:

eqns
vars

after executing the code ans getting the error?

[Yuchi Kang]

Here is the photo for executing the two commands.

I try to use R2018a. It does work and here is the result

sol_sin = 
(N^(1/2)*P*c*x*sin((x*((-E*I*N)^(1/2)*1i + E^(1/2)*I^(1/2)*N^(1/2)))/(E*I))*1i - E^(1/2)*I^(1/2)*P*c*cos((N^(1/2)*x)/(E^(1/2)*I^(1/2)))*1i + E^(1/2)*I^(1/2)*P*c*cos((x*((-E*I*N)^(1/2)*1i + E^(1/2)*I^(1/2)*N^(1/2)))/(E*I))*1i)/(N^(3/2)*l)

[Stephan]

Nice. Problem solved?

[Yuchi Kang]

Yes. Almost done. Thanks. But I have a minor problem with the results. If I want to use this equation k^2=N/E/I to simplify my results. May I use the subs command?

Best regards Yuchi Kang

[Stephan]

yes - this is what subs is for - i would suggest to this before reducing the differential order:

syms y(x) x E I N P c l Dyt(x) k
eqn = diff(y,x,2)+N/E*I*y == -P*c/(l*E*I)*x;
eqn = subs(eqn,(N/E*I),k^2)
[eqns, vars] = reduceDifferentialOrder(eqn,y);
conds = [y(0)==0,Dyt(0)==0];
sol = dsolve(eqns,conds)
sol_sin = rewrite(sol.y,'sin')

which gives:

sol_sin =

- ((P*c*1i)/(2*E*I*k^3*l) + (P*c*(- 1 + k*x*1i)*(sin(k*x)*1i - 2*sin((k*x)/2)^2 + 1)*1i)/(2*E*I*k^3*l))*(sin(k*x)*1i + 2*sin((k*x)/2)^2 - 1) - ((P*c*1i)/(2*E*I*k^3*l) + (P*c*(1 + k*x*1i)*(sin(k*x)*1i + 2*sin((k*x)/2)^2 - 1)*1i)/(2*E*I*k^3*l))*(sin(k*x)*1i - 2*sin((k*x)/2)^2 + 1)

You can also get rid of the imaginary parts of the results by making some assumptions. For example i suspect E to be youngs modul, which should be a real number and positive always:

syms y(x) x E I N P c l Dyt(x) k
eqn = diff(y,x,2)+N/E*I*y == -P*c/(l*E*I)*x;
eqn = subs(eqn,(N/E*I),k^2)
[eqns, vars] = reduceDifferentialOrder(eqn,y);
conds = [y(0)==0,Dyt(0)==0];
assume([x P c l k],'real');
assumeAlso(c>=0 & l>=0 & k>=0);
assumptions
sol = dsolve(eqns,conds)
sol_sin = rewrite(sol.y,'sin')

results in:

sol_sin =

(P*c*sin(k*x))/(E*I*k^3*l) - (P*c*x)/(E*I*k^2*l)

You should check if the assumptions i made are correct in your case and modify them if needed.

[Stephan]

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