60.2.363 problem 940

Internal problem ID [10950]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 940
Date solved : Tuesday, January 28, 2025 at 05:35:43 PM
CAS classification : [[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 63

dsolve(diff(y(x),x) = 1/x*(y(x)*ln(x)*x+x^2*ln(x)-2*x*y(x)-x^2-y(x)^2-y(x)^3+3*x*y(x)^2*ln(x)-3*x^2*ln(x)^2*y(x)+x^3*ln(x)^3)/(-y(x)+x*ln(x)-x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.522 (sec). Leaf size: 57

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

\begin{align*} y(x)\to x \left (\log (x)-\frac {1}{1+\sqrt {-2 x+c_1}}\right ) \\ y(x)\to x \left (\log (x)+\frac {1}{-1+\sqrt {-2 x+c_1}}\right ) \\ y(x)\to x \log (x) \\ \end{align*}