15.4.15 problem 16

Internal problem ID [2928]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 8, page 34
Problem number : 16
Date solved : Monday, January 27, 2025 at 06:56:57 AM
CAS classification : [_exact, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{2} \csc \left (x \right )^{2}+6 y x -2&=\left (2 y \cot \left (x \right )-3 x^{2}\right ) y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.123 (sec). Leaf size: 65

dsolve(y(x)^2*csc(x)^2+6*x*y(x)-2=(2*y(x)*cot(x)-3*x^2)*diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y &= \frac {3 \tan \left (x \right ) x^{2}}{2}-\frac {\sqrt {\tan \left (x \right ) \left (9 \tan \left (x \right ) x^{4}+4 c_1 -8 x \right )}}{2} \\ y &= \frac {3 \tan \left (x \right ) x^{2}}{2}+\frac {\sqrt {\tan \left (x \right ) \left (9 \tan \left (x \right ) x^{4}+4 c_1 -8 x \right )}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 32.781 (sec). Leaf size: 201

DSolve[y[x]^2*Csc[x]^2+6*x*y[x]-2==(2*y[x]*Cot[x]-3*x^2)*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {3}{2} x^2 \tan (x)-\frac {\csc (2 x) \sqrt {-\left (\tan (x) \left (16 \cos ^2(x) \arcsin \left (\sqrt {\sin ^2(x)}\right )-9 x^4 e^{\text {arctanh}(\cos (2 x))}+\cos (2 x) \left (9 x^4 e^{\text {arctanh}(\cos (2 x))}-4 c_1\right )-4 c_1\right )\right )}}{2 \sqrt {\csc (2 x) e^{\text {arctanh}(\cos (2 x))}}} \\ y(x)\to \frac {3}{2} x^2 \tan (x)+\frac {\csc (2 x) \sqrt {-\left (\tan (x) \left (16 \cos ^2(x) \arcsin \left (\sqrt {\sin ^2(x)}\right )-9 x^4 e^{\text {arctanh}(\cos (2 x))}+\cos (2 x) \left (9 x^4 e^{\text {arctanh}(\cos (2 x))}-4 c_1\right )-4 c_1\right )\right )}}{2 \sqrt {\csc (2 x) e^{\text {arctanh}(\cos (2 x))}}} \\ \end{align*}