Internal problem ID [736]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime } x^{2}+\sin \relax (x ) y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = a_{0}, y^{\prime }\relax (0) = a_{1}] \end {align*}
With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
Order:=6; dsolve([diff(y(x),x$2)+x^2*diff(y(x),x)+sin(x)*y(x)=0,y(0) = a__0, D(y)(0) = a__1],y(x),type='series',x=0);
\[ y \relax (x ) = a_{0} +a_{1} x -\frac {1}{6} a_{0} x^{3}-\frac {1}{6} a_{1} x^{4}+\frac {1}{120} a_{0} x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 32
AsymptoticDSolveValue[{y''[x]+x^2*y'[x]+Sin[x]*y[x]==0,{y[0]==a0,y'[0]==a1}},y[x],{x,0,5}]
\[ y(x)\to \frac {\text {a0} x^5}{120}-\frac {\text {a0} x^3}{6}+\text {a0}-\frac {\text {a1} x^4}{6}+\text {a1} x \]