12.5.40 problem 37(a)

Internal problem ID [1664]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 37(a)
Date solved : Monday, January 27, 2025 at 05:22:07 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime }&=\frac {2 y^{2}-y x +2 x^{2}}{y x +2 x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 43

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

\[ y = \operatorname {RootOf}\left (\textit {\_Z}^{4}+c_1 x +16+\left (-3 c_1 x -32\right ) \textit {\_Z} +\left (3 c_1 x +24\right ) \textit {\_Z}^{2}+\left (-c_1 x -8\right ) \textit {\_Z}^{3}\right ) x \]

✓ Solution by Mathematica

Time used: 60.176 (sec). Leaf size: 1913

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

\begin{align*} y(x)\to \frac {1}{12} \left (-\sqrt {9 e^{2 c_1} x^4+6\ 2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+27 e^{2 c_1} x^8}+24 e^{c_1} x^3 \left (-3+\frac {2 \sqrt [3]{2} 3^{2/3} x^2}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}-6 \sqrt {\frac {1}{2} x^2 \left (-8+e^{c_1} x\right ){}^2+4 x^2 \left (-8+e^{c_1} x\right )-\frac {\sqrt [3]{27 e^{2 c_1} x^8+\sqrt {729 e^{4 c_1} x^{16}-6912 e^{3 c_1} x^{15}}}}{3 \sqrt [3]{2}}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}+\frac {e^{c_1} x^4 \left (e^{2 c_1} x^2-12 e^{c_1} x+24\right )}{2 \sqrt {e^{2 c_1} x^4+2 \left (\frac {2}{3}\right )^{2/3} \sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}+e^{c_1} \left (-8 x^3+\frac {16 \sqrt [3]{\frac {2}{3}} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}}}-3 x \left (-8+e^{c_1} x\right )\right ) \\ y(x)\to \frac {1}{12} \left (-\sqrt {9 e^{2 c_1} x^4+6\ 2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+27 e^{2 c_1} x^8}+24 e^{c_1} x^3 \left (-3+\frac {2 \sqrt [3]{2} 3^{2/3} x^2}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}+6 \sqrt {\frac {1}{2} x^2 \left (-8+e^{c_1} x\right ){}^2+4 x^2 \left (-8+e^{c_1} x\right )-\frac {\sqrt [3]{27 e^{2 c_1} x^8+\sqrt {729 e^{4 c_1} x^{16}-6912 e^{3 c_1} x^{15}}}}{3 \sqrt [3]{2}}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}+\frac {e^{c_1} x^4 \left (e^{2 c_1} x^2-12 e^{c_1} x+24\right )}{2 \sqrt {e^{2 c_1} x^4+2 \left (\frac {2}{3}\right )^{2/3} \sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}+e^{c_1} \left (-8 x^3+\frac {16 \sqrt [3]{\frac {2}{3}} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}}}-3 x \left (-8+e^{c_1} x\right )\right ) \\ y(x)\to \frac {1}{12} \left (\sqrt {9 e^{2 c_1} x^4+6\ 2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+27 e^{2 c_1} x^8}+24 e^{c_1} x^3 \left (-3+\frac {2 \sqrt [3]{2} 3^{2/3} x^2}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}-6 \sqrt {\frac {1}{2} x^2 \left (-8+e^{c_1} x\right ){}^2+4 x^2 \left (-8+e^{c_1} x\right )-\frac {\sqrt [3]{27 e^{2 c_1} x^8+\sqrt {729 e^{4 c_1} x^{16}-6912 e^{3 c_1} x^{15}}}}{3 \sqrt [3]{2}}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}-\frac {e^{c_1} x^4 \left (e^{2 c_1} x^2-12 e^{c_1} x+24\right )}{2 \sqrt {e^{2 c_1} x^4+2 \left (\frac {2}{3}\right )^{2/3} \sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}+e^{c_1} \left (-8 x^3+\frac {16 \sqrt [3]{\frac {2}{3}} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}}}-3 x \left (-8+e^{c_1} x\right )\right ) \\ y(x)\to \frac {1}{12} \left (\sqrt {9 e^{2 c_1} x^4+6\ 2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+27 e^{2 c_1} x^8}+24 e^{c_1} x^3 \left (-3+\frac {2 \sqrt [3]{2} 3^{2/3} x^2}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}+6 \sqrt {\frac {1}{2} x^2 \left (-8+e^{c_1} x\right ){}^2+4 x^2 \left (-8+e^{c_1} x\right )-\frac {\sqrt [3]{27 e^{2 c_1} x^8+\sqrt {729 e^{4 c_1} x^{16}-6912 e^{3 c_1} x^{15}}}}{3 \sqrt [3]{2}}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}-\frac {e^{c_1} x^4 \left (e^{2 c_1} x^2-12 e^{c_1} x+24\right )}{2 \sqrt {e^{2 c_1} x^4+2 \left (\frac {2}{3}\right )^{2/3} \sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}+e^{c_1} \left (-8 x^3+\frac {16 \sqrt [3]{\frac {2}{3}} x^5}{\sqrt [3]{\sqrt {3} \sqrt {e^{3 c_1} x^{15} \left (-256+27 e^{c_1} x\right )}+9 e^{2 c_1} x^8}}\right )}}}-3 x \left (-8+e^{c_1} x\right )\right ) \\ \end{align*}