12.6.4 problem 4
Internal
problem
ID
[1683]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Exact
equations.
Section
2.5
Page
79
Problem
number
:
4
Date
solved
:
Monday, January 27, 2025 at 05:27:09 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2 x -2 y^{2}+\left (12 y^{2}-4 y x \right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 426
dsolve((2*x-2*y(x)^2)+(12*y(x)^2-4*x*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-6 x^{5}-6 c_1 \,x^{3}+81 x^{4}+162 c_1 \,x^{2}+81 c_1^{2}}\right )^{{1}/{3}}}{6}+\frac {x^{2}}{6 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-6 x^{5}-6 c_1 \,x^{3}+81 x^{4}+162 c_1 \,x^{2}+81 c_1^{2}}\right )^{{1}/{3}}}+\frac {x}{6} \\
y &= \frac {\left (-\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}+x \right ) \left (i \left (\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}+x \right ) \sqrt {3}+\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}-x \right )}{12 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}}{12}-\frac {x \left (i x \sqrt {3}+x -2 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}\right )}{12 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.898 (sec). Leaf size: 414
DSolve[(2*x-2*y[x]^2)+(12*y[x]^2-4*x*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {x^2}{3\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}-\frac {\sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {x}{6} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) x^2}{6\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}+\frac {x}{6} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{6\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}+\frac {x}{6} \\
\end{align*}