8.3.14 problem 15

Internal problem ID [690]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.4. Separable equations. Page 43
Problem number : 15
Date solved : Monday, January 27, 2025 at 02:58:06 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) = (-1+x)*y(x)^5/x^2/(-y(x)+2*y(x)^3),y(x), singsol=all)
 

\begin{align*} y &= \frac {8 x^{2} 2^{{1}/{3}}-4 x \left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}}+2^{{2}/{3}} \left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{2}/{3}}}{\left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}} \left (6 c_1 x +6 x \ln \left (x \right )+6\right )} \\ y &= -\frac {8 x \left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}}-8 x^{2} \left (i \sqrt {3}-1\right ) 2^{{1}/{3}}+2^{{2}/{3}} \left (1+i \sqrt {3}\right ) \left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{2}/{3}}}{\left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}} \left (12 c_1 x +12 x \ln \left (x \right )+12\right )} \\ y &= \frac {-8 x \left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}}-8 x^{2} \left (1+i \sqrt {3}\right ) 2^{{1}/{3}}+2^{{2}/{3}} \left (i \sqrt {3}-1\right ) \left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{2}/{3}}}{\left (3 x \left (x \ln \left (x \right )+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 x \left (x \ln \left (x \right )+1+\left (c_1 -\frac {4}{3}\right ) x \right ) \left (x \ln \left (x \right )+1+\left (c_1 +\frac {4}{3}\right ) x \right )\right )^{{1}/{3}} \left (12 c_1 x +12 x \ln \left (x \right )+12\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 46.554 (sec). Leaf size: 842

DSolve[D[y[x],x] == (-1+x)*y[x]^5/x^2/(-y[x]+2*y[x]^3),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*}