60.3.431 problem 1437

Internal problem ID [11441]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1437
Date solved : Monday, January 27, 2025 at 11:21:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}+\frac {\sin \left (x \right )^{2} y}{\cos \left (x \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.242 (sec). Leaf size: 29

dsolve(diff(diff(y(x),x),x) = (3*sin(x)^2+1)/cos(x)/sin(x)*diff(y(x),x)+sin(x)^2/cos(x)^2*y(x),y(x), singsol=all)
 

\[ y = c_{1} \cos \left (x \right )^{-\frac {3}{2}+\frac {\sqrt {13}}{2}}+c_{2} \cos \left (x \right )^{-\frac {3}{2}-\frac {\sqrt {13}}{2}} \]

✓ Solution by Mathematica

Time used: 0.286 (sec). Leaf size: 36

DSolve[D[y[x],{x,2}] == Tan[x]^2*y[x] + Csc[x]*Sec[x]*(1 + 3*Sin[x]^2)*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos ^{-\frac {3}{2}-\frac {\sqrt {13}}{2}}(x) \left (c_2 \cos ^{\sqrt {13}}(x)+c_1\right ) \]