1.18 problem 18

Internal problem ID [7334]
Internal file name [OUTPUT/6315_Sunday_June_05_2022_04_39_39_PM_83415849/index.tex]

Book: First order enumerated odes
Section: section 1
Problem number: 18.
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]

\[ \boxed {y^{\prime } c -b y=0} \]

1.18.1 Solving as quadrature ode

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

Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{\frac {b \left (x +c_{1} \right )}{c}}\\ &=c_{1} {\mathrm e}^{\frac {b x}{c}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{\frac {b x}{c}} \\ \end{align*}

Verification of solutions

\[ y = c_{1} {\mathrm e}^{\frac {b x}{c}} \] Verified OK.

1.18.2 Maple step by step solution

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

Maple trace

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

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 13

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

\[ y \left (x \right ) = {\mathrm e}^{\frac {b x}{c}} c_{1} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 21

DSolve[c*y'[x]==b*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 e^{\frac {b x}{c}} \\ y(x)\to 0 \\ \end{align*}