75.23.12 problem 735

Internal problem ID [17209]
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 : 735
Date solved : Tuesday, January 28, 2025 at 09:57:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=-2\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

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

\[ y = -2+2 x -x^{2}+\frac {1}{3} x^{3}-\frac {1}{4} x^{4}+\frac {7}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 34

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

\[ y(x)\to \frac {7 x^5}{60}-\frac {x^4}{4}+\frac {x^3}{3}-x^2+2 x-2 \]