15.25.12 problem 11

Internal problem ID [3399]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 11
Date solved : Monday, January 27, 2025 at 07:35:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=6 \left (-x^{2}+1\right )^{2} \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 26

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

\[ y(x)\to -x^4+3 x^2+c_1 \left (x^2+1\right )+c_2 x \]