69.23.12 problem 735

Internal problem ID [18496]
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 : Thursday, October 02, 2025 at 03:14:22 PM
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*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

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

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

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

ode=D[y[x],{x,2}]-(1+x^2)*y[x]==0; 
ic={y[0]==-2,Derivative[1][y][0] ==2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.227 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x**2 - 1)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): -2, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {x^{5} r{\left (3 \right )}}{20} + C_{2} \left (\frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{20} + 1\right ) + O\left (x^{6}\right ) \]