7.3.15 problem 15
Internal
problem
ID
[55]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.4
(separable
equations).
Problems
at
page
43
Problem
number
:
15
Date
solved
:
Friday, February 07, 2025 at 07:47:12 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {\left (x -1\right ) y^{5}}{x^{2} \left (2 y^{3}-y\right )} \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 840
dsolve(diff(y(x),x)= ((x-1)*y(x)^5)/(x^2*(2*y(x)^3-y(x))),y(x), singsol=all)
\begin{align*}
y &= \frac {8 x^{2} 2^{{1}/{3}}-4 x \left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}}+2^{{2}/{3}} \left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{2}/{3}}}{\left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}} \left (6 c_1 x +6 \ln \left (x \right ) x +6\right )} \\
y &= -\frac {8 x \left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}}-8 \left (i \sqrt {3}-1\right ) x^{2} 2^{{1}/{3}}+\left (1+i \sqrt {3}\right ) 2^{{2}/{3}} \left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{2}/{3}}}{\left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}} \left (12 c_1 x +12 \ln \left (x \right ) x +12\right )} \\
y &= \frac {-8 x \left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}}-8 \left (1+i \sqrt {3}\right ) x^{2} 2^{{1}/{3}}+\left (i \sqrt {3}-1\right ) 2^{{2}/{3}} \left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{2}/{3}}}{\left (3 x \left (\ln \left (x \right ) x +c_1 x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_1 \,x^{2}+x \right ) \ln \left (x \right )+\left (9 c_1^{2}-32\right ) x^{2}+18 c_1 x}+9 \left (\ln \left (x \right ) x +1+\left (c_1 -\frac {4}{3}\right ) x \right ) x \left (\ln \left (x \right ) x +1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}} \left (12 c_1 x +12 \ln \left (x \right ) x +12\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 44.027 (sec). Leaf size: 842
DSolve[D[y[x],x]== ((x-1)*y[x]^5)/(x^2*(2*y[x]^3-y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\frac {8 \sqrt [3]{2} x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}+4 x}{6 (x \log (x)+c_1 x+1)} \\
y(x)\to \frac {\frac {8 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}-8 x}{12 (x \log (x)+c_1 x+1)} \\
y(x)\to \frac {\frac {8 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{16 x^3-9 x^3 \log ^2(x)-9 c_1{}^2 x^3-18 c_1 x^2+3 \sqrt {x^2 (x \log (x)+c_1 x+1){}^2 \left (9 x^2 \log ^2(x)+\left (-32+9 c_1{}^2\right ) x^2+18 c_1 x+18 x (1+c_1 x) \log (x)+9\right )}-18 x^2 (1+c_1 x) \log (x)-9 x}-8 x}{12 (x \log (x)+c_1 x+1)} \\
y(x)\to 0 \\
\end{align*}