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 : Friday, February 07, 2025 at 07:59:04 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]==f[a*x+b*y[x]+c],y[x],x,IncludeSingularSolutions -> True]
 

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