76.47.4 problem Ex. 4

Internal problem ID [20264]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 102
Problem number : Ex. 4
Date solved : Thursday, October 02, 2025 at 05:38:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }+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: 39

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

\[ y = \left (1-x^{2}+\frac {2}{3} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {17}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 40

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

\[ y(x)\to c_2 \left (\frac {17 x^5}{40}-\frac {x^3}{2}+x\right )+c_1 \left (\frac {2 x^4}{3}-x^2+1\right ) \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 29

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

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