50.6.12 problem 4

Internal problem ID [7904]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.8. Integrating Factors. Page 32
Problem number : 4
Date solved : Monday, January 27, 2025 at 03:31:41 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.689 (sec). Leaf size: 176

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

\begin{align*} y &= \frac {x^{2} \left (-\cot \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) c_{1} +\csc \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) x \right )}{c_{1}} \\ y &= \frac {\left (\cot \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) c_{1} +\csc \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) x \right ) x^{2}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.146 (sec). Leaf size: 36

DSolve[D[y[x],x]==2*y[x]/x+x^3/y[x]+x*Tan[y[x]/x^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [3 \log (x)-\log \left (y(x) \sin \left (\frac {y(x)}{x^2}\right )+x^2 \cos \left (\frac {y(x)}{x^2}\right )\right )=c_1,y(x)\right ] \]