2.6.9 Problem 9 (c)

Internal problem ID [18494]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 33. Problems at page 91
Problem number : 9 (c)
Date solved : Thursday, March 13, 2025 at 12:07:55 PM
CAS classification : [_quadrature]

Solve

$$\begin{array}{rl}\sqrt{1+v^{\prime }}& =\frac{{\mathrm{e}}^u}{2}\end{array}$$

Solved as first order quadrature ode

Time used: 0.080 (sec)

Since the ode has the form $v^{\prime }=f(u)$, then we only need to integrate $f(u)$.

$$\begin{array}{rl}\int dv& =\int \frac{{\mathrm{e}}^{2u}}{4}-1du\\ v& =-u+\frac{{\mathrm{e}}^{2u}}{8}+c_1\end{array}$$

Figure 2.67: Slope field $\sqrt{1+v^{\prime }}=\frac{{\mathrm{e}}^u}{2}$

Summary of solutions found

$$\begin{array}{rl}v& =-u+\frac{{\mathrm{e}}^{2u}}{8}+c_1\end{array}$$

✓ Maple. Time used: 0.009 (sec). Leaf size: 17

ode:=(1+diff(v(u),u))^(1/2) = 1/2*exp(u); 
dsolve(ode,v(u), singsol=all);
 

$$v=\frac{{\mathrm{e}}^{2u}}{8}-\ln \left({\mathrm{e}}^u\right)+c_1$$

Maple trace

`Methods for first order ODEs: 
-> Solving 1st order ODE of high degree, 1st attempt 
trying 1st order WeierstrassP solution for high degree ODE 
trying 1st order WeierstrassPPrime solution for high degree ODE 
trying 1st order JacobiSN solution for high degree ODE 
trying 1st order ODE linearizable_by_differentiation 
trying differential order: 1; missing variables 
<- differential order: 1; missing  y(x)  successful`
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & \sqrt{1+v^{\prime }}=\frac{{\mathrm{e}}^u}{2}\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & v^{\prime }\\ \bullet & & \text{Solve for the highest derivative}\\ & & v^{\prime }=\frac{{\left({\mathrm{e}}^u\right)}^2}{4}-1\\ \bullet & & \text{Integrate both sides with respect to}u\\ & & \int v^{\prime }du=\int (\frac{{\left({\mathrm{e}}^u\right)}^2}{4}-1)du+\mathit{C}\mathit{1}\\ \bullet & & \text{Evaluate integral}\\ & & v=-u+\frac{{\left({\mathrm{e}}^u\right)}^2}{8}+\mathit{C}\mathit{1}\\ \bullet & & \text{Solve for}v\\ & & v=-u+\frac{{\left({\mathrm{e}}^u\right)}^2}{8}+\mathit{C}\mathit{1}\end{array}$$

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 20

ode=Sqrt[1+D[v[u],u]]==Exp[u]/2; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

$$v(u)\to -u+\frac{e^{2u}}{8}+c_1$$

✓ Sympy. Time used: 0.192 (sec). Leaf size: 12

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq(sqrt(Derivative(v(u), u) + 1) - exp(u)/2,0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

$$v\left(u\right)=C_1-u+\frac{e^{2u}}{8}$$