"dsolve" gives some extra terms in the solution

I am facing problem in using dsolve. Similar issue was also discussed in below link. I tried the suggestion given by Star Strider in question (https://in.mathworks.com/matlabcentral/answers/326248-dsolve-returns-imaginary-components-when-it-shouldnt-how-to-fix-this), but it does not work in my case or maybe I am doing something wrong. Here is my code:

syms Vs(x) w l C phi Vcc Vr
q= 1/(w*sqrt(l*C));
ode = w*w*l*C*diff(Vs,x,2)+Vs==Vcc-Vr*sin(x+phi); % non-homogenous second order DE
Vs = dsolve(ode,x,'IgnoreAnalyticConstraints',1);
Vs= simplify(Vs);
pretty(Vs)

This gives the solution

Vr sin(phi + x) - Vcc - C5 exp(#1) - C6 exp(-#1) + C Vcc l (w^2)  + C C5 l (w^2)  exp(#1) + C C6 l (w^2)  exp(-#1)
----------------------------------------------------------------------------------------------------------
                                                     C l (w^2)  - 1
         x sqrt(-C l)
   #1 == ------------
             C l w

which after some simplication with hand gives the following:

             Vs(x)= (q^2/(1-q^2))*Vr*Sin(phi+x)-Vcc-C5*cosqx-i*C5*Sinqx-C6*cosqx+i*C6*sinqx

If the DE is solved by hand it gives the following:

             V(x)= C1*cos(q*x) + C2*sin(q*x) + Vcc + (q^2)/(1-q^2)*Vr*sin(phi+x)

The expected solution as calculated by hand does not contain any imaginary part. Moreover the solution given by dsolve does not fully match with the expected one. I don't undestand where I am doing wrong. Kindly help me to debug this.

[darova]

Did you try to plot it?

[Mir Nazir]

No I haven't since I have too many unknowns.

[darova]

What C1 and C2 represents? I suppose they are initial conditions (positions and veloctity)?

The same as C5 and C6

[Mir Nazir]

C1 and C2 are constants of general solution of the differential equation in question. C1 and C2 will be calculated using initial conditions afterwards. First I wanted to have a general solution(homogenous + particular solution) of the equation. Yes, C1,C2 are same as C5, C4.The latter being generated by Matlab itself.

[darova]

I think there is something wrong in your hand calculations

V(x)= C1cosqxC2sinqx+Vcc-(q^2-1)/(q^2)*Vrsin(phi+x)

I wanted to calculate C1 and C2 and failed because when i tried to substitute x=0 all constants disappear

V(0) = Vcc-(q^2-1)/(q^2)*Vrsin(phi+0)       % wrong?

[Mir Nazir]

I apologise. There is + sign in between c1cosqx and c2 sinqx. It is a typographical error I have corrected it in the poriginal post.

Answer 1

[Star Strider]

Tell simplify to keep simplifying until it either meets the iteration limit or cannot simplify the expression further:

Vs= simplify(Vs, 'Steps',500);
pretty(Vs)

producing:

Vs =
Vcc + C1*exp((x*(-C*l)^(1/2))/(C*l*w)) + C2*exp(-(x*(-C*l)^(1/2))/(C*l*w)) + (Vr*sin(phi + x))/(C*l*w^2 - 1)

with the LaTeX version being:

[Mir Nazir]

Actually I don't have any complex part in my expected answer which is from an IEEE transaction article and also coincides with my hand calculations. I have attached the screenshot of the equation I am trying to solve along with its answer.

[Mir Nazir]

Any suggestions?

Answer 2

[darova]

Here is comparison of solutions. See script inside

[Mir Nazir]

Thank you so much for pointing out the error. I am sincerly sorry. I have re-calculated the results and have accordingly edited the post. This solution does coincide with that given by dsolve using your script (plot and script attached). But the problem of imaginary terms in the general solution still persists.

[Mir Nazir]

@darova looking forward for your valuable suggestions.

[darova]

The best i can

-C5*cosqx-i*C5*Sinqx-C6*cosqx+i*C6*sinqx
i*sin(qx)*(C6-C5) - cos(qx)*(C6+C5)
% C1 = i*(C6-C5)
% C2 = -(C6+C5)
C1*sin(qx) + C2*cos(qx)

[Mir Nazir]

Thank you darova. Is my code correct?

[darova]

It looks similar now. I think so

[Mir Nazir]

But as per explanation above the value for C1 is complex then but actually it is not. I am attaching the solutions for c1 and c2 as well

[darova]

Can't explain this. Looks like phenomena

Did you try to solve equations for C5 and C6 in original form?

-C5*cosqx-i*C5*Sinqx-C6*cosqx+i*C6*sinqx

Maybe during some calculations complex i disappears somewhere?

[Mir Nazir]

Thanks for the reply. I will try that.