10.5.3 problem 3
Internal
problem
ID
[1195]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Section
2.6.
Page
100
Problem
number
:
3
Date
solved
:
Monday, January 27, 2025 at 04:43:49 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2+3 x^{2}-2 y x +\left (3-x^{2}+6 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 368
dsolve(2+3*x^2-2*x*y(x)+(3-x^2+6*y(x)^2)*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{2}/{3}}+6 x^{2}-18}{6 \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}} \\
y &= \frac {\left (-1-i \sqrt {3}\right ) \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}}{12}+\frac {\left (x^{2}-3\right ) \left (i \sqrt {3}-1\right )}{2 \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{2}/{3}}}{6}+\left (-1-i \sqrt {3}\right ) \left (x^{2}-3\right )}{2 \left (-54 x^{3}-54 c_1 -108 x +6 \sqrt {75 x^{6}+162 c_1 \,x^{3}+378 x^{4}+81 c_1^{2}+324 c_1 x +162 x^{2}+162}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 8.618 (sec). Leaf size: 421
DSolve[2+3*x^2-2*x*y[x]+(3-x^2+6*y[x]^2)*D[y[x],x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {x^2-3}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}}-\frac {\sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}}{6^{2/3}} \\
y(x)\to \frac {\sqrt [3]{6} \left (1+i \sqrt {3}\right ) \left (x^2-3\right )+\left (1-i \sqrt {3}\right ) \left (9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}} \\
y(x)\to \frac {\sqrt [3]{6} \left (1-i \sqrt {3}\right ) \left (x^2-3\right )+\left (1+i \sqrt {3}\right ) \left (9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}} \\
\end{align*}