1.10 problem 10

Internal problem ID [7326]
Internal file name [OUTPUT/6307_Sunday_June_05_2022_04_39_24_PM_88182402/index.tex]

Book: First order enumerated odes
Section: section 1
Problem number: 10.
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]

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

1.10.1 Solving as quadrature ode

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

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

Summary

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

Verification of solutions

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

1.10.2 Maple step by step solution

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

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: 10

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

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

✓ Solution by Mathematica

Time used: 0.03 (sec). Leaf size: 18

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

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