60.3.347 problem 1353

Internal problem ID [11357]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1353
Date solved : Monday, January 27, 2025 at 11:17:50 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 {y}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 65

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

\[ y = \frac {-c_{1} \sqrt {2}\, \sqrt {\pi }\, \left (x^{4}+2 x^{2}-1\right ) \operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right )+2 c_{1} \left (x^{3}-x \right ) {\mathrm e}^{\frac {1}{2 x^{2}}}+c_{2} \left (x^{4}+2 x^{2}-1\right )}{x} \]

✓ Solution by Mathematica

Time used: 1.370 (sec). Leaf size: 66

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

\[ y(x)\to \frac {e^3 \left (x^4+2 x^2-1\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2 K[1]^2}-6} K[1]^4}{\left (K[1]^4+2 K[1]^2-1\right )^2}dK[1]+c_1\right )}{x} \]