60.2.162 problem 738
Internal
problem
ID
[10749]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
738
Date
solved
:
Monday, January 27, 2025 at 09:39:53 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y^{\prime }&=\frac {2 a}{-x^{2} y+2 a y^{4} x^{2}-16 a^{2} x y^{2}+32 a^{3}} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 964
dsolve(diff(y(x),x) = 2*a/(-x^2*y(x)+2*a*y(x)^4*x^2-16*a^2*x*y(x)^2+32*a^3),y(x), singsol=all)
\begin{align*}
y &= \frac {192 c_{1}^{2} a^{3} x +x^{2}-x \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{1}/{3}}+\left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{2}/{3}}}{12 c_{1} a x \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{2}/{3}}}{24}+8 x \left (-\frac {\left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{1}/{3}}}{96}+\left (a^{3} c_{1}^{2}+\frac {x}{192}\right ) \left (i \sqrt {3}-1\right )\right )}{c_{1} a x \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{2}/{3}}}{24}+8 \left (-\frac {\left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{1}/{3}}}{96}+\left (-i \sqrt {3}-1\right ) \left (a^{3} c_{1}^{2}+\frac {x}{192}\right )\right ) x}{c_{1} a x \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.432 (sec). Leaf size: 335
DSolve[D[y[x],x] == (2*a)/(32*a^3 - x^2*y[x] - 16*a^2*x*y[x]^2 + 2*a*x^2*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\left (\frac {4 a y(x)^3+4 a y(x)^2+1}{-16 y(x) a^2-16 a^2+K[1] \left (4 a y(x)^3+4 a y(x)^2+1\right )}-\frac {y(x)^2}{K[1] y(x)^2-4 a}\right )dK[1]+\int _1^{y(x)}\left (-\frac {2 x K[2]}{x K[2]^2-4 a}-\int _1^x\left (\frac {2 K[1] K[2]^3}{\left (K[1] K[2]^2-4 a\right )^2}-\frac {2 K[2]}{K[1] K[2]^2-4 a}+\frac {12 a K[2]^2+8 a K[2]}{-16 K[2] a^2-16 a^2+K[1] \left (4 a K[2]^3+4 a K[2]^2+1\right )}-\frac {\left (4 a K[2]^3+4 a K[2]^2+1\right ) \left (K[1] \left (12 a K[2]^2+8 a K[2]\right )-16 a^2\right )}{\left (-16 K[2] a^2-16 a^2+K[1] \left (4 a K[2]^3+4 a K[2]^2+1\right )\right )^2}\right )dK[1]-\frac {4 \left (4 a^2-3 x K[2]^2 a-2 x K[2] a\right )}{4 a x K[2]^3+4 a x K[2]^2-16 a^2 K[2]-16 a^2+x}\right )dK[2]=c_1,y(x)\right ]
\]