1
$\begingroup$

How do I find the series solution to the following non-linear second order differential equation up to and including the $x^4$ term and plot the solution curve using Mathematica?

$y''= x + y^2$ (with given initial conditions: $y(0)=0$ and $y'(0)=0$)

$\endgroup$
  • 1
    $\begingroup$ What have you tried so far? $\endgroup$ – Anne Nov 20 '17 at 4:14
  • $\begingroup$ I tried inputting the following code to solve and plot the solution: a = DSolve[{y''[x] == x + (y[x])^2, y[0] == 0, y'[0] == 0}, y[x], x]; Plot[y[x] /. a, {x, -2, 2}], but it did not compute. I have no clue about how to find the series solution either. $\endgroup$ – Saeed Mohanna Nov 20 '17 at 4:21
4
$\begingroup$

The series solution is just $\frac{x^3}{6}$ (for these initial conditions)

findSeriesSolution[t_,nTerms_]:=Module[
  {pt=0,u,ode,s0,s1,ic,eq,sol,roots},
  ic={u[0]->0,u'[0]->0};
  ode=u''[t]-t-u[t]^2;
  s0=Series[ode,{t,pt,nTerms}];
  s0=s0/.ic;
  roots=Solve@LogicalExpand[s0==0];
  s1=Series[u[t],{t,pt,nTerms+2}];
  sol=Normal[s1]/.ic/.roots[[1]]
]

And

seriesSol=findSeriesSolution[x,5]

Mathematica graphics

Plot[x^3/6,{x,0,2}]

Mathematica graphics

Compare to NDSolve

ClearAll[u,x]
sol=NDSolve[{u''[x]==x+u[x]^2 ,u[0]==0,u'[0]==0},u,{x,0,2}];
Plot[Evaluate[u[x]/.sol],{x,0,2}]

Mathematica graphics

Verify also using Maple

Mathematica graphics

If you want general series solution for any initial conditions, then replace the line

ic = {u[0] ->0, u'[0] -> 0};

with

ic = {u[0] -> C[1], u'[0] -> C[2]};

In the above function., And now

seriesSol=findSeriesSolution[x,5]

$$ \frac{\left(20 c_2 c_1^3+6 c_1^2+5 c_2^3\right) x^7}{1260}+\frac{1}{360} \left(5 c_1^4+10 c_2^2 c_1+4 c_2\right) x^6+\frac{1}{60} \left(5 c_2 c_1^2+c_1\right) x^5+\frac{1}{12} \left(c_1^3+c_2^2\right) x^4+\frac{1}{6} \left(2 c_1 c_2+1\right) x^3+\frac{1}{2} c_1^2 x^2+c_2 x+c_1 $$

You can see that for C[1]=0 and C[2]=0 the above gives $\frac{x^3}{6}$ since all other terms are zero.

$\endgroup$
  • 1
    $\begingroup$ Ohh ok, thank you so much! $\endgroup$ – Saeed Mohanna Nov 20 '17 at 4:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.