56.4.12 problem 12

Internal problem ID [8901]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 12
Date solved : Monday, January 27, 2025 at 05:18:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 217

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{3861}+\frac {x^{9/2}}{810}+\frac {x^{7/2}}{630}+\frac {2 x^{5/2}}{75}+\frac {4 x^{3/2}}{15}-2 \sqrt {x}+\frac {2}{\sqrt {x}}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \left (\frac {x^6}{20790}+\frac {37 x^5}{69300}-\frac {17 x^4}{5040}-\frac {x^3}{84}-\frac {x}{3}-\frac {1}{x}+\log (x)\right ) \]