23.1.134 problem 138

Internal problem ID [4741]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 138
Date solved : Tuesday, September 30, 2025 at 08:20:52 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.101 (sec). Leaf size: 262

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

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

✓ Sympy. Time used: 1.106 (sec). Leaf size: 66

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

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