Internal problem ID [8188]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 608.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\sqrt {y}}{\sqrt {y}+F \left (\frac {x -y}{\sqrt {y}}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.229 (sec). Leaf size: 40
dsolve(diff(y(x),x) = y(x)^(1/2)/(y(x)^(1/2)+F((x-y(x))/y(x)^(1/2))),y(x), singsol=all)
\[ \frac {\ln \left (y \relax (x )\right )}{2}-\left (\int _{}^{\frac {x}{\sqrt {y \relax (x )}}-\sqrt {y \relax (x )}}\frac {1}{2 F \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.301 (sec). Leaf size: 274
DSolve[y'[x] == Sqrt[y[x]]/(F[(x - y[x])/Sqrt[y[x]]] + Sqrt[y[x]]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {F\left (\frac {x-K[2]}{\sqrt {K[2]}}\right )}{x \sqrt {K[2]}}-\int _1^x-\frac {-\frac {F\left (\frac {K[1]-K[2]}{\sqrt {K[2]}}\right )}{\sqrt {K[2]}}-2 \left (-\frac {K[1]-K[2]}{2 K[2]^{3/2}}-\frac {1}{\sqrt {K[2]}}\right ) \sqrt {K[2]} F'\left (\frac {K[1]-K[2]}{\sqrt {K[2]}}\right )-1}{\left (-2 \sqrt {K[2]} F\left (\frac {K[1]-K[2]}{\sqrt {K[2]}}\right )+K[1]-K[2]\right )^2}dK[1]+\frac {2 F\left (\frac {x-K[2]}{\sqrt {K[2]}}\right )^2+\sqrt {K[2]} F\left (\frac {x-K[2]}{\sqrt {K[2]}}\right )+x}{x \left (-x+K[2]+2 F\left (\frac {x-K[2]}{\sqrt {K[2]}}\right ) \sqrt {K[2]}\right )}\right )dK[2]+\int _1^x\frac {1}{-2 \sqrt {y(x)} F\left (\frac {K[1]-y(x)}{\sqrt {y(x)}}\right )+K[1]-y(x)}dK[1]=c_1,y(x)\right ] \]