60.3.399 problem 1405

Internal problem ID [11409]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1405
Date solved : Monday, January 27, 2025 at 11:19:49 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 {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 41

dsolve(diff(diff(y(x),x),x) = (2*x^2+1)/x^3*diff(y(x),x)-1/4*(a*x^4+10*x^2+1)/x^6*y(x),y(x), singsol=all)
 

\[ y = x^{{3}/{2}} {\mathrm e}^{-\frac {1}{4 x^{2}}} \left (c_{1} x^{\frac {\sqrt {-a +9}}{2}}+c_{2} x^{-\frac {\sqrt {-a +9}}{2}}\right ) \]

✓ Solution by Mathematica

Time used: 0.127 (sec). Leaf size: 74

DSolve[D[y[x],{x,2}] == -1/4*((1 + 10*x^2 + a*x^4)*y[x])/x^6 + ((1 + 2*x^2)*D[y[x],x])/x^3,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-\frac {1}{4 x^2}-\frac {3}{2}} x^{\frac {3}{2}-\frac {\sqrt {9-a}}{2}} \left (c_2 x^{\sqrt {9-a}}+\sqrt {9-a} c_1\right )}{\sqrt {9-a}} \]