84.31.6 problem 19.7

Internal problem ID [22304]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 19. Power series solutions about an ordinary point. Solved problems. Page 98
Problem number : 19.7
Date solved : Thursday, October 02, 2025 at 08:37:19 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-2 y x&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ y^{\prime }\left (2\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

\[ y = 1+6 \left (x -2\right )+2 \left (x -2\right )^{2}+\frac {13}{3} \left (x -2\right )^{3}+\frac {5}{3} \left (x -2\right )^{4}+\frac {16}{15} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 44

ode=D[y[x],{x,2}]-2*x*y[x]==0; 
ic={y[2]==1,Derivative[1][y][2] ==6}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to \frac {16}{15} (x-2)^5+\frac {5}{3} (x-2)^4+\frac {13}{3} (x-2)^3+2 (x-2)^2+6 (x-2)+1 \]

✓ Sympy. Time used: 0.297 (sec). Leaf size: 51

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

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