2.29 problem 29

Internal problem ID [507]
Internal file name [OUTPUT/507_Sunday_June_05_2022_01_42_40_AM_35330923/index.tex]

Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Section 2.2. Page 48
Problem number: 29.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y^{\prime }-\frac {b +a y}{d +c y}=0} \]

2.29.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {c y +d}{a y +b}d y &= x +c_{1}\\ \frac {c y}{a}+\frac {\left (a d -b c \right ) \ln \left (a y +b \right )}{a^{2}}&=x +c_{1} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} a^{2}+x \,a^{2}-\left (-\operatorname {LambertW}\left (-\frac {c \,{\mathrm e}^{\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}}}{-a d +b c}\right )+\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}\right ) a d +\left (-\operatorname {LambertW}\left (-\frac {c \,{\mathrm e}^{\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}}}{-a d +b c}\right )+\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}\right ) b c}{a c} \\ \end{align*}

Verification of solutions

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

2.29.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {b +a y}{d +c y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {b +a y}{d +c y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (d +c y\right )}{b +a y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (d +c y\right )}{b +a y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {c y}{a}+\frac {\left (a d -b c \right ) \ln \left (b +a y\right )}{a^{2}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {c_{1} a^{2}+x \,a^{2}-\left (-\mathit {LambertW}\left (-\frac {c \,{\mathrm e}^{\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}}}{-a d +b c}\right )+\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}\right ) a d +\left (-\mathit {LambertW}\left (-\frac {c \,{\mathrm e}^{\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}}}{-a d +b c}\right )+\frac {c_{1} a^{2}+x \,a^{2}+b c}{a d -b c}\right ) b c}{a c} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 61

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

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

✓ Solution by Mathematica

Time used: 16.166 (sec). Leaf size: 83

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

\begin{align*} y(x)\to \frac {-b c+(a d-b c) W\left (-\frac {c \left (e^{-1-\frac {a^2 (x+c_1)}{b c}}\right ){}^{\frac {b c}{b c-a d}}}{b c-a d}\right )}{a c} \\ y(x)\to -\frac {b}{a} \\ \end{align*}