78.5.25 problem 5

Internal problem ID [18168]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 9 (Integrating Factors). Problems at page 80
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:35:56 AM
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: 0.500 (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 (-c_1 \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 )+x \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 )\right )}{c_1} \\ y &= \frac {x^{2} \left (c_1 \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 )+x \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 )\right )}{c_1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.098 (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]
 

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