60.3.425 problem 1431

Internal problem ID [11435]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1431
Date solved : Tuesday, January 28, 2025 at 06:06:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.397 (sec). Leaf size: 30

dsolve(diff(diff(y(x),x),x) = cos(2*x)/sin(2*x)*diff(y(x),x)-2*y(x),y(x), singsol=all)
 

\[ y = \sin \left (2 x \right )^{{3}/{4}} \left (c_{1} \operatorname {LegendreP}\left (\frac {1}{4}, \frac {3}{4}, \cos \left (2 x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {1}{4}, \frac {3}{4}, \cos \left (2 x \right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.985 (sec). Leaf size: 119

DSolve[D[y[x],{x,2}] == -2*y[x] + Cot[2*x]*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

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