1.13 problem 13

Internal problem ID [7329]
Internal file name [OUTPUT/6310_Sunday_June_05_2022_04_39_31_PM_47734585/index.tex]

Book: First order enumerated odes
Section: section 1
Problem number: 13.
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]

\[ \boxed {y^{\prime } c=a} \]

1.13.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \frac {a}{c}\,\mathop {\mathrm {d}x}}\\ &= \frac {x a}{c}+c_{1} \end {align*}

Summary

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

Verification of solutions

\[ y = \frac {x a}{c}+c_{1} \] Verified OK.

1.13.2 Maple step by step solution

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

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: 0.015 (sec). Leaf size: 12

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

\[ y \left (x \right ) = \frac {a x}{c}+c_{1} \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 14

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

\[ y(x)\to \frac {a x}{c}+c_1 \]