60.2.184 problem 760
Internal
problem
ID
[10771]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
760
Date
solved
:
Tuesday, January 28, 2025 at 05:13:23 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {\left (x y^{2}+1\right )^{3}}{x^{4} \left (x y^{2}+1+x \right ) y} \end{align*}
✓ Solution by Maple
Time used: 0.121 (sec). Leaf size: 174
dsolve(diff(y(x),x) = (x*y(x)^2+1)^3/x^4/(x*y(x)^2+1+x)/y(x),y(x), singsol=all)
\begin{align*}
y &= -\frac {\sqrt {2}\, \sqrt {-x \left (2+\left (1+i\right ) x \right )}}{2 x} \\
y &= \frac {\sqrt {2}\, \sqrt {-x \left (2+\left (1+i\right ) x \right )}}{2 x} \\
y &= -\frac {\sqrt {2}\, \sqrt {x \left (-2+\left (-1+i\right ) x \right )}}{2 x} \\
y &= \frac {\sqrt {2}\, \sqrt {x \left (-2+\left (-1+i\right ) x \right )}}{2 x} \\
-\frac {\ln \left (2 x^{2} y^{4}+\left (2 x^{2}+4 x \right ) y^{2}+x^{2}+2 x +2\right )}{10}+\frac {\arctan \left (2 y^{4} x +\left (2+2 x \right ) y^{2}+x +1\right )}{10}+\frac {1}{2 x}+\frac {\ln \left (x y^{2}-x +1\right )}{5}-\frac {\arctan \left (2 y^{2}+1\right )}{10}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.768 (sec). Leaf size: 426
DSolve[D[y[x],x] == (1 + x*y[x]^2)^3/(x^4*y[x]*(1 + x + x*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {4 x K[2]}{5 \left (x K[2]^2-x+1\right )}-\int _1^x\left (-\frac {4 K[2]}{5 \left (K[1] K[2]^2-K[1]+1\right )}+\frac {4 K[1] \left (K[2]^2-1\right ) K[2]}{5 \left (K[1] K[2]^2-K[1]+1\right )^2}+\frac {16 K[1] K[2]^3+8 K[1] K[2]+8 K[2]}{5 \left (2 K[1]^2 K[2]^4+2 K[1]^2 K[2]^2+4 K[1] K[2]^2+K[1]^2+2 K[1]+2\right )}-\frac {\left (8 K[1]^2 K[2]^3+4 K[1]^2 K[2]+8 K[1] K[2]\right ) \left (4 K[1] K[2]^4+4 K[1] K[2]^2+4 K[2]^2+2 K[1]+1\right )}{5 \left (2 K[1]^2 K[2]^4+2 K[1]^2 K[2]^2+4 K[1] K[2]^2+K[1]^2+2 K[1]+2\right )^2}\right )dK[1]+\frac {2 \left (4 x^2 K[2]^3+3 x^2 K[2]+4 x K[2]\right )}{5 \left (2 x^2 K[2]^4+2 x^2 K[2]^2+4 x K[2]^2+x^2+2 x+2\right )}\right )dK[2]+\int _1^x\left (-\frac {2 \left (y(x)^2-1\right )}{5 \left (K[1] y(x)^2-K[1]+1\right )}+\frac {4 K[1] y(x)^4+4 K[1] y(x)^2+4 y(x)^2+2 K[1]+1}{5 \left (2 K[1]^2 y(x)^4+2 K[1]^2 y(x)^2+4 K[1] y(x)^2+K[1]^2+2 K[1]+2\right )}+\frac {1}{K[1]^2}\right )dK[1]=c_1,y(x)\right ]
\]