38.2.32 problem 32
Internal
problem
ID
[6461]
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
Date
solved
:
Monday, January 27, 2025 at 02:04:56 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime }&=\frac {2 y x +y^{2}}{x^{2}+2 y x} \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 352
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 &= \frac {12^{{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 )^{{1}/{3}}}{6 c_1}+\frac {x 12^{{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 )^{{1}/{3}}}+x \\
y &= \frac {-\frac {2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_1 \,x^{2}-4 x}{c_1}}+9 x \right ) c_1^{2}\right )}^{{2}/{3}}}{6}+x c_1 \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 )}^{{1}/{3}}+2^{{1}/{3}} \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right )\right )}{2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_1 \,x^{2}-4 x}{c_1}}+9 x \right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= -\frac {-\frac {2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_1 \,x^{2}-4 x}{c_1}}+9 x \right ) c_1^{2}\right )}^{{2}/{3}}}{6}+x \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 )}^{{1}/{3}}+2^{{1}/{3}} \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right )\right ) 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 )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 51.692 (sec). Leaf size: 404
DSolve[D[y[x],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*}