75.23.13 problem 736

Internal problem ID [17210]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.1 Integration of differential equation in series. Power series. Exercises page 171
Problem number : 736
Date solved : Tuesday, January 28, 2025 at 09:57:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=x^{2} y-y^{\prime } \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

Order:=6; 
dsolve([diff(y(x),x$2)=x^2*y(x)-diff(y(x),x),y(0) = 1, D(y)(0) = 0],y(x),type='series',x=0);
 

\[ y = 1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 19

AsymptoticDSolveValue[{D[y[x],{x,2}]==x^2*y[x]-D[y[x],x],{y[0]==1,Derivative[1][y][0] ==0}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to -\frac {x^5}{60}+\frac {x^4}{12}+1 \]