76.1.37 problem 37

Internal problem ID [17336]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 37
Date solved : Tuesday, January 28, 2025 at 10:01:36 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\frac {a y+b}{c y+d} \end{align*}

✓ Solution by Maple

Time used: 0.085 (sec). Leaf size: 61

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

\[ y = \frac {\left (a d -b c \right ) \operatorname {LambertW}\left (\frac {c \,{\mathrm e}^{\frac {\left (x +c_{1} \right ) a^{2}+b c}{a d -b c}}}{a d -b c}\right )-b c}{a c} \]

✓ Solution by Mathematica

Time used: 0.156 (sec). Leaf size: 44

DSolve[D[y[x],x]==(a*y[x]+b)/(c*y[x]+d),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {d+c K[1]}{b+a K[1]}dK[1]\&\right ][x+c_1] \\ y(x)\to -\frac {b}{a} \\ \end{align*}