60.2.37 problem 613

Internal problem ID [10624]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 613
Date solved : Tuesday, January 28, 2025 at 04:56:18 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 34

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

\begin{align*} y &= x \left (\ln \left (x \right )+\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right )\right ) \\ y &= \left (\ln \left (x \right )+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right )\right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.208 (sec). Leaf size: 226

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

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x F\left (\frac {K[2]-x \log (x)}{x}\right ) \int _1^x\left (-\frac {K[2] F''\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )^2 K[1]^3}-\frac {F''\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )^2 K[1]^2}+\frac {1}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right ) K[1]^2}\right )dK[1]+1}{x F\left (\frac {K[2]-x \log (x)}{x}\right )}dK[2]+\int _1^x\left (\frac {y(x)}{F\left (\frac {y(x)-K[1] \log (K[1])}{K[1]}\right ) K[1]^2}+\frac {1}{F\left (\frac {y(x)-K[1] \log (K[1])}{K[1]}\right ) K[1]}+1\right )dK[1]=c_1,y(x)\right ] \]