82.47.4 problem Ex. 4

Internal problem ID [18979]
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 : Tuesday, January 28, 2025 at 12:41:18 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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

AsymptoticDSolveValue[(2*x^2+1)*D[y[x],{x,2}]+x*D[y[x],x]+2*y[x]==0,y[x],{x,0,"6"-1}]
 

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