2.28 problem 21

Internal problem ID [12926]
Internal file name [OUTPUT/11579_Tuesday_November_07_2023_11_27_18_PM_8602688/index.tex]

Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number: 21.
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]

\[ \boxed {v^{\prime }-\frac {K -v}{R C}=0} \]

2.28.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {R C}{K -v}d v &= t +c_{1}\\ -R C \ln \left (K -v \right )&=t +c_{1} \end {align*}

Solving for \(v\) gives these solutions \begin {align*} v_1&=-{\mathrm e}^{-\frac {t +c_{1}}{R C}}+K\\ &=-\frac {{\mathrm e}^{-\frac {t}{R C}}}{c_{1}}+K \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} v &= -\frac {{\mathrm e}^{-\frac {t}{R C}}}{c_{1}}+K \\ \end{align*}

Verification of solutions

\[ v = -\frac {{\mathrm e}^{-\frac {t}{R C}}}{c_{1}}+K \] Verified OK.

2.28.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & v^{\prime }-\frac {K -v}{R C}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & v^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & v^{\prime }=\frac {K -v}{R C} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {v^{\prime }}{K -v}=\frac {1}{R C} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {v^{\prime }}{K -v}d t =\int \frac {1}{R C}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (K -v\right )=\frac {t}{R C}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \\ {} & {} & v=-{\mathrm e}^{-\frac {c_{1} R C +t}{R C}}+K \end {array} \]

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: 18

dsolve(diff(v(t),t)=(K-v(t))/(R*C),v(t), singsol=all)
 

\[ v \left (t \right ) = K +c_{1} {\mathrm e}^{-\frac {t}{R C}} \]

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 26

DSolve[v'[t]==(k-v[t])/(r*c),v[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} v(t)\to k+c_1 e^{-\frac {t}{c r}} \\ v(t)\to k \\ \end{align*}