60.3.76 problem 1078

Internal problem ID [11086]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1078
Date solved : Monday, January 27, 2025 at 10:45:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 33

dsolve(diff(diff(y(x),x),x)+f(x)*diff(y(x),x)+(1/4*f(x)^2+1/2*diff(f(x),x)+a)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {\left (\int fd x \right )}{2}} \left (c_{1} \sinh \left (\sqrt {-a}\, x \right )+c_{2} \cosh \left (\sqrt {-a}\, x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.122 (sec). Leaf size: 69

DSolve[y[x]*(a + f[x]^2/4 + Derivative[1][f][x]/2) + f[x]*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (2 \sqrt {a} c_1-i c_2 e^{2 i \sqrt {a} x}\right ) \exp \left (-\frac {1}{2} \int _1^xf(K[1])dK[1]-i \sqrt {a} x\right )}{2 \sqrt {a}} \]