78.5.21 problem 4 (j)

Internal problem ID [18164]
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 : 4 (j)
Date solved : Tuesday, January 28, 2025 at 11:35:46 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} -y+x y^{\prime }&=x^{2} y^{4} \left (x y^{\prime }+y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 23

dsolve(x*diff(y(x),x)-y(x)= x^2*y(x)^4*(x*diff(y(x),x)+y(x)),y(x), singsol=all)
 

\[ y = \operatorname {RootOf}\left (x^{6} \textit {\_Z}^{4}+3-{\mathrm e}^{\frac {3 c_1}{2}} \textit {\_Z} \right ) x \]

✓ Solution by Mathematica

Time used: 60.122 (sec). Leaf size: 1141

DSolve[x*D[y[x],x]-y[x] == x^2*y[x]^4*(x*D[y[x],x]+y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}-\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}-\frac {6 c_1}{x^3 \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}}} \\ y(x)\to \frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}-\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}-\frac {6 c_1}{x^3 \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}-\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}+\frac {6 c_1}{x^3 \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}+\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}-\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}+\frac {6 c_1}{x^3 \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}+\frac {\sqrt [3]{9 c_1{}^2 x^3+\sqrt {-256 x^{12}+81 c_1{}^4 x^6}}}{\sqrt [3]{2} x^3}}}} \\ \end{align*}