60.2.1 problem 577

Internal problem ID [10588]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 577
Date solved : Monday, January 27, 2025 at 09:16:55 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 28

dsolve(diff(y(x),x) = F(y(x)/(x+a)),y(x), singsol=all)
 

\[ y = -\operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (-\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +\ln \left (a +x \right )+c_{1} \right ) \left (a +x \right ) \]

✓ Solution by Mathematica

Time used: 0.205 (sec). Leaf size: 243

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

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