60.2.147 problem 723
Internal
problem
ID
[10734]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
723
Date
solved
:
Monday, January 27, 2025 at 09:36:36 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y^{\prime }&=\frac {2 a}{y+2 a y^{4}-16 a^{2} x y^{2}+32 a^{3} x^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.060 (sec). Leaf size: 692
dsolve(diff(y(x),x) = 2*a/(y(x)+2*a*y(x)^4-16*a^2*x*y(x)^2+32*a^3*x^2),y(x), singsol=all)
\begin{align*}
y &= \frac {{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{1}/{3}}}{6 a}+\frac {8 a^{2} \left (a \,c_{1}^{2}+3 x \right )}{3 {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{1}/{3}}}+\frac {2 c_{1} a}{3} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{2}/{3}}}{12}+\frac {4 \left (\frac {c_{1} {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{1}/{3}}}{2}+a \left (i \sqrt {3}-1\right ) \left (a \,c_{1}^{2}+3 x \right )\right ) a^{2}}{3}}{a {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{2}/{3}}}{12}+\frac {4 a^{2} \left (\frac {c_{1} {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{1}/{3}}}{2}+\left (-i \sqrt {3}-1\right ) a \left (a \,c_{1}^{2}+3 x \right )\right )}{3}}{a {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 17.548 (sec). Leaf size: 672
DSolve[D[y[x],x] == (2*a)/(32*a^3*x^2 + y[x] - 16*a^2*x*y[x]^2 + 2*a*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{12 \sqrt [3]{2} a}-\frac {8 a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {4 \left (1+i \sqrt {3}\right ) a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {4 \left (1-i \sqrt {3}\right ) a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3} \\
\end{align*}