60.1.253 problem 254

Internal problem ID [10267]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 254
Date solved : Monday, January 27, 2025 at 06:42:39 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} x \left (y x -2\right ) y^{\prime }+x^{2} y^{3}+x y^{2}-2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 59

dsolve(x*(x*y(x)-2)*diff(y(x),x)+x^2*y(x)^3+x*y(x)^2-2*y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {-1+\sqrt {1-4 \ln \left (x \right )+4 c_{1}}}{2 x \left (-\ln \left (x \right )+c_{1} \right )} \\ y &= \frac {1+\sqrt {1-4 \ln \left (x \right )+4 c_{1}}}{2 \left (\ln \left (x \right )-c_{1} \right ) x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.188 (sec). Leaf size: 86

DSolve[x*(x*y[x]-2)*D[y[x],x]+x^2*y[x]^3+x*y[x]^2-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2}{x+\sqrt {-\frac {1}{x^3}} x^2 \sqrt {-x (-4 \log (x)+1+4 c_1)}} \\ y(x)\to \frac {2}{x+\left (-\frac {1}{x^3}\right )^{3/2} x^5 \sqrt {-x (-4 \log (x)+1+4 c_1)}} \\ y(x)\to 0 \\ \end{align*}