60.1.83 problem 84

Internal problem ID [10097]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 84
Date solved : Sunday, March 30, 2025 at 03:15:49 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }-f \left (a x +b y\right )&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 37

ode:=diff(y(x),x)-f(a*x+b*y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_a} b \right ) b +a}d \textit {\_a} b -x +c_1 \right ) b -a x}{b} \]

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 248

ode=D[y[x],x] - f[a*x + b*y[x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.570 (sec). Leaf size: 66

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
f = Function("f") 
ode = Eq(-f(a*x + b*y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} a^{2} + b^{2} \left (C_{1} - a \int \limits ^{- C_{2} - \frac {a x}{b}} \frac {f{\left (- r b \right )}}{a + b f{\left (- r b \right )}}\, dr + b \int \limits ^{- C_{2} - \frac {a x}{b}} \frac {1}{a + b f{\left (- r b \right )}}\, dr + x\right )}{a b} \]