2.1.24 Problem 24

Internal problem ID [9008]
Book : First order enumerated odes
Section : section 1
Problem number : 24
Date solved : Wednesday, March 05, 2025 at 07:14:27 AM
CAS classification : [_quadrature]

Solve

$$\begin{array}{rl}a\sin \left(x\right)yx{y}^{\prime }& =0\end{array}$$

Factoring the ode gives these factors

$$\begin{array}{}\text{(1)}& y& =0\text{(2)}& y^{\prime }& =0\end{array}$$

Now each of the above equations is solved in turn.

Solving equation (1)

Solving for $y$ from

$$\begin{array}{r}y=0\end{array}$$

Solving gives $y=0$

Solving equation (2)

Solved as first order quadrature ode

Time used: 0.013 (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 2.19: Slope field $a\sin \left(x\right)yx{y}^{\prime }=0$

Summary of solutions found

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

Solved as first order homogeneous class D2 ode

Time used: 0.091 (sec)

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

$$\begin{array}{r}a\sin \left(x\right)u\left(x\right)x^2(u^{\prime }\left(x\right)x+u\left(x\right))=0\end{array}$$

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 2.20: Slope field $a\sin \left(x\right)yx{y}^{\prime }=0$

Summary of solutions found

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 9

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

$$\begin{array}{rl}y& =0\\ y& =c_1\end{array}$$

Maple trace

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

Maple step by step

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 12

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

$$\begin{array}{r}y(x)\to 0\\ y(x)\to c_1\end{array}$$

✓ Sympy. Time used: 0.133 (sec). Leaf size: 3

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

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