60.2.32 problem 608

Internal problem ID [10619]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 608
Date solved : Monday, January 27, 2025 at 09:18:14 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.213 (sec). Leaf size: 38

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)
 

\[ \frac {\ln \left (y\right )}{2}-\int _{}^{\frac {x -y}{\sqrt {y}}}\frac {1}{2 F \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} -c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.248 (sec). Leaf size: 274

DSolve[D[y[x],x] == Sqrt[y[x]]/(F[(x - y[x])/Sqrt[y[x]]] + Sqrt[y[x]]),y[x],x,IncludeSingularSolutions -> True]
 

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