Internal problem ID [5118]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page
1068
Problem number: 32.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y^{\prime }-\frac {2 x y+y^{2}}{x^{2}+2 x y}=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 356
dsolve(diff(y(x),x)=(2*x*y(x)+y(x)^2)/(x^2+2*x*y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}+x \\ y \left (x \right ) &= \frac {-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{6}+\left (2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x c_{1}}{2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= -\frac {-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{6}+\left (-2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x c_{1}}{2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 56.42 (sec). Leaf size: 404
DSolve[y'[x]==(2*x*y[x]+y[x]^2)/(x^2+2*x*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{\frac {2}{3}} e^{c_1} x}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}{\sqrt [3]{2} 3^{2/3}}+x \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) e^{c_1} x}{2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-27 e^{c_1} x^2}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}{2 \sqrt [3]{2} 3^{2/3}}+x \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) e^{c_1} x}{2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-27 e^{c_1} x^2}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}{2 \sqrt [3]{2} 3^{2/3}}+x \\ \end{align*}