34.6.12 problem 12

Internal problem ID [6086]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number : 12
Date solved : Monday, January 27, 2025 at 01:35:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 43

dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-25/4)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {-3 c_2 \left (i x -\frac {1}{3} x^{2}+1\right ) {\mathrm e}^{-i x}+3 \,{\mathrm e}^{i x} \left (i x +\frac {1}{3} x^{2}-1\right ) c_1}{x^{{5}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.071 (sec). Leaf size: 59

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-25/4)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left (\left (-c_2 x^2+3 c_1 x+3 c_2\right ) \cos (x)+\left (c_1 \left (x^2-3\right )+3 c_2 x\right ) \sin (x)\right )}{x^{5/2}} \]