87.15.17 problem 17

Internal problem ID [23565]
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 : 17
Date solved : Thursday, October 02, 2025 at 09:42:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {1}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{2}+\frac {3}{2} x^{3}-\frac {13}{24} x^{4}-\frac {13}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^5}{40}-\frac {x^4}{8}+\frac {x^3}{6}+1\right )+c_2 \left (-\frac {13 x^5}{20}-\frac {13 x^4}{24}+\frac {3 x^3}{2}-\frac {3 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.303 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + (2*x**2 + 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)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{4}}{8} + \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (- \frac {13 x^{3}}{24} + \frac {3 x^{2}}{2} - \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]