7.5.55 problem 55

Internal problem ID [159]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 55
Date solved : Saturday, March 29, 2025 at 04:37:25 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

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

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

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

✓ Mathematica. Time used: 0.235 (sec). Leaf size: 262

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

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

✓ Sympy. Time used: 1.756 (sec). Leaf size: 66

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

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