33.6.4 problem Problem 27.37

Internal problem ID [7849]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 27. Power series solutions of linear DE with variable coefficients. Supplementary Problems. page 274
Problem number : Problem 27.37
Date solved : Tuesday, September 30, 2025 at 05:06:18 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 37

Order:=6; 
ode:=diff(diff(y(x),x),x)-2*x*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+x^{2}+\frac {1}{2} x^{4}\right ) y \left (0\right )+\left (x +\frac {2}{3} x^{3}+\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

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

\[ y(x)\to c_2 \left (\frac {4 x^5}{15}+\frac {2 x^3}{3}+x\right )+c_1 \left (\frac {x^4}{2}+x^2+1\right ) \]

✓ Sympy. Time used: 0.207 (sec). Leaf size: 29

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

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