2.1.32 Problem 32

Internal problem ID [9016]
Book : First order enumerated odes
Section : section 1
Problem number : 32
Date solved : Wednesday, March 05, 2025 at 07:14:45 AM
CAS classification : [_quadrature]

Solve

$$\begin{array}{rl}5y^{\prime }& =0\end{array}$$

Solved as first order quadrature ode

Time used: 0.014 (sec)

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

$$\begin{array}{rl}\int dy& =\int 0dx+c_1\\ y& =c_1\end{array}$$

Figure 2.32: Slope field $5y^{\prime }=0$

Summary of solutions found

$$\begin{array}{rl}y& =c_1\end{array}$$

Solved as first order homogeneous class D2 ode

Time used: 0.056 (sec)

Applying change of variables $y=u\left(x\right)x$, then the ode becomes

$$\begin{array}{r}5u^{\prime }\left(x\right)x+5u\left(x\right)=0\end{array}$$

Which is now solved The ode

$$\begin{array}{}\text{(1)}& u^{\prime }\left(x\right)=-\frac{u\left(x\right)}{x}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{u}^{\prime }\left(x\right)& =-\frac{u\left(x\right)}{x}\\ & =f(x)g(u)\end{array}$$

Where

$$\begin{array}{rl}f(x)& =-\frac{1}{x}\\ g(u)& =u\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(u)}du& =\int f(x)dx\\ \int \frac{1}{u}du& =\int -\frac{1}{x}dx\end{array}$$

$$\ln \left(u\left(x\right)\right)=\ln \left(\frac{1}{x}\right)+c_1$$

Taking the exponential of both sides the solution becomes

$$u\left(x\right)=\frac{c_1}{x}$$

Converting $u\left(x\right)=\frac{c_1}{x}$ back to $y$ gives

$$\begin{array}{r}y=c_1\end{array}$$

Figure 2.33: Slope field $5y^{\prime }=0$

Summary of solutions found

$$\begin{array}{rl}y& =c_1\end{array}$$

✓ Maple. Time used: 0.001 (sec). Leaf size: 5

ode:=5*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=c_1$$

Maple trace

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

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & 5\frac{d}{dx}y\left(x\right)=0\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Separate variables}\\ & & \frac{d}{dx}y\left(x\right)=0\\ \bullet & & \text{Integrate both sides with respect to}x\\ & & \int \left(\frac{d}{dx}y\left(x\right)\right)dx=\int 0dx+\mathit{C}\mathit{1}\\ \bullet & & \text{Evaluate integral}\\ & & y\left(x\right)=\mathit{C}\mathit{1}\\ \bullet & & \text{Solve for}y\left(x\right)\\ & & y\left(x\right)=\mathit{C}\mathit{1}\end{array}$$

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 7

ode=5*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to c_1$$

✓ Sympy. Time used: 0.036 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

$$y\left(x\right)=C_1$$