Internal problem ID [8856]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1276.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } x^{2}+4 x y^{\prime }-\left (4 x^{2}+1\right ) y-4 \sqrt {x^{3}}\, {\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 31
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(4*x^2+1)*y(x)-4*(x^3)^(1/2)*exp(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\sinh \relax (x ) c_{2}}{\sqrt {x}}+\frac {\cosh \relax (x ) c_{1}}{\sqrt {x}}+\frac {\sqrt {x^{3}}\, {\mathrm e}^{x}}{2 x} \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 53
DSolve[-4*E^x*Sqrt[x^3] - (1 + 4*x^2)*y[x] + 4*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^x x (2 x-1)}{4 \sqrt {x^3}}+\frac {c_1 e^{-x}}{\sqrt {x}}+\frac {c_2 e^x}{2 \sqrt {x}} \\ \end{align*}