40.3.25 problem 25 (j)

Internal problem ID [6629]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 5. Equations of first order and first degree (Exact equations). Supplemetary problems. Page 33
Problem number : 25 (j)
Date solved : Monday, January 27, 2025 at 02:16:20 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y+x \left (x^{2} y-1\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.142 (sec). Leaf size: 673

dsolve(y(x)+x*(x^2*y(x)-1)*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 57.790 (sec). Leaf size: 452

DSolve[y[x]+x*(x^2*y[x]-1)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{-6 c_1} \sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}+\frac {e^{6 c_1}}{\sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}}+1}{2 x^2} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) e^{-6 c_1} \sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1}}{\sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}}+2}{4 x^2} \\ y(x)\to \frac {-\left (\left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}\right )+\frac {i \left (\sqrt {3}+i\right ) e^{6 c_1}}{\sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}}+2}{4 x^2} \\ y(x)\to 0 \\ y(x)\to \frac {3}{2 x^2} \\ \end{align*}