Internal problem ID [8383]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 803.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\textit {\_F1} \left (y^{2}-2 \ln \relax (x )\right )}{\sqrt {y^{2}}\, x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 65
dsolve(diff(y(x),x) = _F1(y(x)^2-2*ln(x))/(y(x)^2)^(1/2)/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {2 \ln \relax (x )+2 \RootOf \left (\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} \right )+c_{1}\right )} \\ y \relax (x ) = -\sqrt {2 \ln \relax (x )+2 \RootOf \left (\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} \right )+c_{1}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.407 (sec). Leaf size: 603
DSolve[y'[x] == F1[-2*Log[x] + y[x]^2]/(x*Sqrt[y[x]^2]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]^2} \text {F1}\left (K[2]^2-2 \log (x)\right )}{\left (\text {F1}\left (K[2]^2-2 \log (x)\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (x)\right )+1\right )}+\frac {K[2]}{\left (\text {F1}\left (K[2]^2-2 \log (x)\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (x)\right )+1\right )}-\int _1^x\left (\frac {2 K[2] \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right )^2 \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 K[2] \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right )^2 K[1]}-\frac {4 K[2] \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 \sqrt {K[2]^2} \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right )^2 \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 \sqrt {K[2]^2} \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right )^2 K[1]}-\frac {2 \sqrt {K[2]^2} \text {F1}'\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}\right )dK[1]\right )dK[2]+\int _1^x\left (-\frac {\text {F1}\left (y(x)^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )+1\right ) K[1]}-\frac {\sqrt {y(x)^2} \text {F1}\left (y(x)^2-2 \log (K[1])\right )}{\left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )+1\right ) K[1] y(x)}\right )dK[1]=c_1,y(x)\right ] \]