Internal problem ID [9273]
Internal file name [OUTPUT/8209_Monday_June_06_2022_02_17_39_AM_21376205/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 940.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+\ln \left (x \right ) x -x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} \ln \left (x \right )^{3}-3 x^{2} \ln \left (x \right )^{2} y-y^{\prime } \ln \left (x \right ) x^{2}+3 x y^{2} \ln \left (x \right )+y^{\prime } y x +y^{\prime } x^{2}-y^{3}+y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-y^{2}-2 x y-x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{3} \ln \left (x \right )^{3}+3 x^{2} \ln \left (x \right )^{2} y-3 x y^{2} \ln \left (x \right )+y^{3}-y \ln \left (x \right ) x -x^{2} \ln \left (x \right )+y^{2}+2 x y+x^{2}}{-x^{2} \ln \left (x \right )+x y+x^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (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 \left (x \right ) &= \frac {x \left (\ln \left (x \right ) \sqrt {-2 x +c_{1}}-\ln \left (x \right )+1\right )}{\sqrt {-2 x +c_{1}}-1} \\ y \left (x \right ) &= \frac {x \left (\ln \left (x \right ) \sqrt {-2 x +c_{1}}+\ln \left (x \right )-1\right )}{\sqrt {-2 x +c_{1}}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.45 (sec). Leaf size: 57
DSolve[y'[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*}