60.3.29 problem 1029

Internal problem ID [11039]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1029
Date solved : Tuesday, January 28, 2025 at 05:40:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 22

dsolve(diff(diff(y(x),x),x)-(f(x)^2+diff(f(x),x))*y(x)=0,y(x), singsol=all)
 

\[ y = \left (\int {\mathrm e}^{-2 \left (\int fd x \right )}d x +c_{1} \right ) {\mathrm e}^{\int fd x} c_{2} \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 58

DSolve[-(y[x]*(f[x]^2 + Derivative[1][f][x])) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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