56.4.6 problem 6

Internal problem ID [8895]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 6
Date solved : Monday, January 27, 2025 at 05:18:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 45

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{63} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 160

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

\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{1980}-\frac {x^{7/2}}{35}-\frac {2 x^{3/2}}{3}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (\frac {x^5}{840}+\frac {x^3}{18}+x\right ) \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \]