87.15.20 problem 20

Internal problem ID [23568]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 115
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:42:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+x \right ) y^{\prime \prime }+3 y^{\prime }-2 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 49

Order:=6; 
ode:=(1+x)*diff(diff(y(x),x),x)+3*diff(y(x),x)-2*x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{6} x^{4}-\frac {1}{5} x^{5}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{2}+2 x^{3}-\frac {5}{2} x^{4}+\frac {31}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 54

ode=(x+1)*D[y[x],{x,2}]+3*D[y[x],x]-2*x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{5}+\frac {x^4}{6}+1\right )+c_2 \left (\frac {31 x^5}{10}-\frac {5 x^4}{2}+2 x^3-\frac {3 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.301 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2*y(x) + (x + 1)*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = - \frac {5 x^{4} r{\left (3 \right )}}{4} + \frac {3 x^{5} r{\left (3 \right )}}{2} + C_{2} \left (- \frac {x^{5}}{5} + \frac {x^{4}}{6} + 1\right ) + C_{1} x \left (\frac {x^{4}}{10} - \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]