75.20.27 problem 666

Internal problem ID [17161]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 666
Date solved : Tuesday, January 28, 2025 at 09:54:49 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }&=\cos \left (x \right ) \cot \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 15

dsolve(diff(y(x),x$2)+tan(x)*diff(y(x),x)=cos(x)*cot(x),y(x), singsol=all)
 

\[ y = c_{2} +\sin \left (x \right ) \left (-1+\ln \left (\sin \left (x \right )\right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.352 (sec). Leaf size: 287

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

\[ y(x)\to \exp \left (\int _1^{\cos (x)}-\frac {1}{2 K[1]-2 K[1]^3}dK[1]-\frac {1}{2} \int _1^{\cos (x)}\frac {1}{K[2] \left (K[2]^2-1\right )}dK[2]\right ) \left (\int _1^{\cos (x)}-\frac {\exp \left (\int _1^{K[4]}-\frac {1}{2 K[1]-2 K[1]^3}dK[1]+\frac {1}{2} \int _1^{K[4]}\frac {1}{K[2] \left (K[2]^2-1\right )}dK[2]\right ) K[4]^2 \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}-\frac {1}{2 K[1]-2 K[1]^3}dK[1]\right )dK[3]}{\left (1-K[4]^2\right )^{3/2}}dK[4]+\int _1^{\cos (x)}\exp \left (-2 \int _1^{K[3]}-\frac {1}{2 K[1]-2 K[1]^3}dK[1]\right )dK[3] \left (\int _1^{\cos (x)}\frac {\exp \left (\int _1^{K[5]}-\frac {1}{2 K[1]-2 K[1]^3}dK[1]+\frac {1}{2} \int _1^{K[5]}\frac {1}{K[2] \left (K[2]^2-1\right )}dK[2]\right ) K[5]^2}{\left (1-K[5]^2\right )^{3/2}}dK[5]+c_2\right )+c_1\right ) \]