60.2.16 problem 592

Internal problem ID [10603]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 592
Date solved : Tuesday, January 28, 2025 at 04:55:43 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=\frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x} \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 33

dsolve(diff(y(x),x) = 1/5*(6*x^3+5*x^(1/2)+5*F(y(x)-2/5*x^3-2*x^(1/2)))/x,y(x), singsol=all)
 

\[ \int _{\textit {\_b}}^{y}\frac {1}{F \left (\textit {\_a} -\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}d \textit {\_a} -\ln \left (x \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.320 (sec). Leaf size: 241

DSolve[D[y[x],x] == (Sqrt[x] + (6*x^3)/5 + F[-2*Sqrt[x] - (2*x^3)/5 + y[x]])/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (-\frac {2 x^3}{5}-2 \sqrt {x}+K[2]\right ) \int _1^x\left (-\frac {6 F''\left (-\frac {2}{5} K[1]^3-2 \sqrt {K[1]}+K[2]\right ) K[1]^2}{5 F\left (-\frac {2}{5} K[1]^3-2 \sqrt {K[1]}+K[2]\right )^2}-\frac {F''\left (-\frac {2}{5} K[1]^3-2 \sqrt {K[1]}+K[2]\right )}{F\left (-\frac {2}{5} K[1]^3-2 \sqrt {K[1]}+K[2]\right )^2 \sqrt {K[1]}}\right )dK[1]+1}{F\left (-\frac {2 x^3}{5}-2 \sqrt {x}+K[2]\right )}dK[2]+\int _1^x\left (\frac {6 K[1]^2}{5 F\left (-\frac {2}{5} K[1]^3-2 \sqrt {K[1]}+y(x)\right )}+\frac {1}{F\left (-\frac {2}{5} K[1]^3-2 \sqrt {K[1]}+y(x)\right ) \sqrt {K[1]}}+\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ] \]