10.6.19 problem 19
Internal
problem
ID
[1236]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Miscellaneous
problems,
end
of
chapter
2.
Page
133
Problem
number
:
19
Date
solved
:
Monday, January 27, 2025 at 04:47:03 AM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {3 x^{2}-2 y-y^{3}}{2 x +3 x y^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 405
dsolve(diff(y(x),x) = (3*x^2-2*y(x)-y(x)^3)/(2*x+3*x*y(x)^2),y(x), singsol=all)
\begin{align*}
y &= -\frac {12^{{1}/{3}} \left (x^{2} 12^{{1}/{3}}-\frac {{\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{2}/{3}}}{2}\right )}{3 {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (2 i 2^{{2}/{3}} 3^{{5}/{6}} x^{2}+i \sqrt {3}\, {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{2}/{3}}-2 x^{2} 2^{{2}/{3}} 3^{{1}/{3}}+{\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{2}/{3}}\right )}{12 x {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (2 \,2^{{2}/{3}} x^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )+\left (i \sqrt {3}-1\right ) {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{2}/{3}}\right )}{12 {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_1 \,x^{3}+27 c_1^{2}+32 x^{2}}-9 c_1 \right ) x^{2}\right )}^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.104 (sec). Leaf size: 358
DSolve[D[y[x],x] == (3*x^2-2*y[x]-y[x]^3)/(2*x+3*x*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}{3 \sqrt [3]{2} x}-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) x}{\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}{6 \sqrt [3]{2} x} \\
y(x)\to \frac {\sqrt [3]{2} \left (1-i \sqrt {3}\right ) x}{\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}{6 \sqrt [3]{2} x} \\
\end{align*}