60.3.385 problem 1391

Internal problem ID [11395]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1391
Date solved : Monday, January 27, 2025 at 11:19:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 20

dsolve(diff(diff(y(x),x),x) = 1/x*(7*a*x^2+5)/(a*x^2+1)*diff(y(x),x)-(15*a*x^2+5)/x^2/(a*x^2+1)*y(x),y(x), singsol=all)
 

\[ y = c_{1} x^{5}+2 a c_{2} x^{3}+c_{2} x \]

✓ Solution by Mathematica

Time used: 0.348 (sec). Leaf size: 122

DSolve[D[y[x],{x,2}] == -(((5 + 15*a*x^2)*y[x])/(x^2*(1 + a*x^2))) + ((5 + 7*a*x^2)*D[y[x],x])/(x*(1 + a*x^2)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3 a K[1]^2+5}{2 a K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^x-\frac {7 a K[2]^2+5}{a K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 a K[1]^2+5}{2 a K[1]^3+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]