56.4.31 problem 27

Internal problem ID [8920]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 27
Date solved : Monday, January 27, 2025 at 05:19:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*(2-x^2)*diff(y(x), x$2) - x*(3+4*x^2)*diff(y(x), x) + (2-2*x^2)*y(x) = 0,y(x),type='series',x=0);
 

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {15}{8} x^{2}+\frac {189}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {6}{7} x^{2}+\frac {45}{77} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 50

AsymptoticDSolveValue[x^2*(2-x^2)*D[y[x],{x,2}] - x*(3+4*x^2)*D[y[x],x] + (2-2*x^2)*y[x] == 0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {45 x^4}{77}+\frac {6 x^2}{7}+1\right ) x^2+c_2 \left (\frac {189 x^4}{128}+\frac {15 x^2}{8}+1\right ) \sqrt {x} \]