2.1.41 Problem 41

Internal problem ID [10299]
Book : First order enumerated odes
Section : section 1
Problem number : 41
Date solved : Wednesday, April 29, 2026 at 07:51:33 AM
CAS classification : [_quadrature]

2.1.41.1 Solved by factoring the differential equation

Time used: 0.136 (sec)

\begin{align*} x y^{\prime } y&=0 \\ \end{align*}

Writing the ode as

\begin{align*} \left (y^{\prime }\right )\left (y\right )&=0 \end{align*}

Therefore we need to solve the following equations

\begin{align*} \tag{1} y^{\prime } &= 0 \\ \tag{2} y &= 0 \\ \end{align*}

Now each of the above equations is solved in turn.

Solving equation (1)

Entering first order ode quadrature solverSince the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\). Integrating gives

\begin{align*} y &= c_1 \end{align*}

Figure 2.30: Phase plot for \(y^{\prime } = 0\)

Solving equation (2)

Entering zero order ode solverSolving for \(y\) from

\begin{align*} y = 0 \end{align*}

Solving gives

\begin{align*} y &= 0 \\ \end{align*}

Summary of solutions found

\begin{align*} y &= 0 \\ y &= c_1 \\ \end{align*}

2.1.41.2 ✓ Maple. Time used: 0.000 (sec). Leaf size: 9

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

\begin{align*} y &= 0 \\ y &= c_1 \\ \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int 0d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \end {array} \]

2.1.41.3 ✓ Mathematica. Time used: 0.002 (sec). Leaf size: 12

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

\begin{align*} y(x)&\to 0\\ y(x)&\to c_1 \end{align*}

2.1.41.4 ✓ Sympy. Time used: 0.141 (sec). Leaf size: 3

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

\[ y{\left (x \right )} = C_{1} \]

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', 'nth_algebraic', 'separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_power_series', 'lie_group', 'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral', '1st_linear_Integral', 'Bernoulli_Integral')