64.3.18 problem 24

Internal problem ID [13289]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 24
Date solved : Tuesday, January 28, 2025 at 05:16:04 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

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

✓ Solution by Maple

Time used: 0.316 (sec). Leaf size: 28

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

\[ y = \cot \left (\operatorname {RootOf}\left (4 c_{1} \sin \left (\textit {\_Z} \right )^{4}-4 \textit {\_Z} \sin \left (\textit {\_Z} \right )^{4}-x^{4}\right )\right ) x \]

✓ Solution by Mathematica

Time used: 0.129 (sec). Leaf size: 116

DSolve[(y[x]+x*(x^2+y[x]^2)^2)+(y[x]*(x^2+y[x]^2)^2-x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (K[2]^3+x^2 K[2]-\int _1^x\left (-\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}+2 K[1] K[2]+\frac {1}{K[1]^2+K[2]^2}\right )dK[1]-\frac {x}{x^2+K[2]^2}\right )dK[2]+\int _1^x\left (K[1]^3+y(x)^2 K[1]+\frac {y(x)}{K[1]^2+y(x)^2}\right )dK[1]=c_1,y(x)\right ] \]