34.6.10 problem 10

Internal problem ID [6084]
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 : 10
Date solved : Monday, January 27, 2025 at 01:35:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 17

dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(x^2+1/4)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_1 \sinh \left (x \right )+c_2 \cosh \left (x \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 32

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]-(x^2+1/4)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-x} \left (c_2 e^{2 x}+2 c_1\right )}{2 \sqrt {x}} \]