59.1.308 problem 313

Internal problem ID [9480]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 313
Date solved : Monday, January 27, 2025 at 06:03:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 28

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

\[ y = c_{2} \left (x +1\right ) \sqrt {x \left (x +2\right )}+2 \left (x^{2}+2 x +\frac {1}{2}\right ) c_{1} \]

✓ Solution by Mathematica

Time used: 1.805 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \cosh \left (4 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x}{x+2}}}\right )\right )+i c_2 \sinh \left (4 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x}{x+2}}}\right )\right ) \]