60.3.348 problem 1354

Internal problem ID [11358]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1354
Date solved : Monday, January 27, 2025 at 11:17:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.130 (sec). Leaf size: 33

dsolve(diff(diff(y(x),x),x) = 1/x^3*(2*x^2-1)*diff(y(x),x)-2/x^4*y(x),y(x), singsol=all)
 

\[ y = \frac {c_{2} x^{5} \operatorname {hypergeom}\left (\left [-\frac {5}{2}\right ], \left [-\frac {1}{2}\right ], \frac {1}{2 x^{2}}\right )+5 c_{1} x^{2}-c_{1}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.456 (sec). Leaf size: 65

DSolve[D[y[x],{x,2}] == (-2*y[x])/x^4 + ((-1 + 2*x^2)*D[y[x],x])/x^3,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{9/2} \left (5 x^2-1\right ) \left (c_2 \int _1^x\frac {25 e^{\frac {1}{2 K[1]^2}-9} K[1]^6}{\left (1-5 K[1]^2\right )^2}dK[1]+c_1\right )}{5 x^2} \]