60.3.77 problem 1079

Internal problem ID [11087]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1079
Date solved : Monday, January 27, 2025 at 10:45:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 33

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

\[ y = c_{1} {\mathrm e}^{i \sqrt {b}\, \left (\int f^{a}d x \right )}+c_{2} {\mathrm e}^{-i \sqrt {b}\, \left (\int f^{a}d x \right )} \]

✓ Solution by Mathematica

Time used: 0.512 (sec). Leaf size: 139

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

\begin{align*} y(x)\to \frac {1}{2} \left (\exp \left (\int _1^x-i \sqrt {b} f(K[1])^adK[1]+c_2\right )-2 c_1 \exp \left (-\int _1^x-i \sqrt {b} f(K[1])^adK[1]-c_2\right )\right ) \\ y(x)\to \frac {1}{2} \left (\exp \left (\int _1^xi \sqrt {b} f(K[2])^adK[2]+c_2\right )-2 c_1 \exp \left (-\int _1^xi \sqrt {b} f(K[2])^adK[2]-c_2\right )\right ) \\ \end{align*}