2.1.46 Problem 46

Internal problem ID [10304]
Book : First order enumerated odes
Section : section 1
Problem number : 46
Date solved : Thursday, November 27, 2025 at 10:32:43 AM
CAS classification : [_quadrature]

Solved by factoring the differential equation

Time used: 0.044 (sec)

Solve

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

Writing the ode as

\begin{align*} \left (y\right )\left ({y^{\prime }}^{2}\right )&=0 \end{align*}

Therefore we need to solve the following equations

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

Now each of the above equations is solved in turn.

Solving equation (1)

Solving for \(y\) from

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

Solving gives

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

Solving equation (2)

Solve Solving for the derivative gives these ODE’s to solve

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

Now each of the above is solved separately.

Solving Eq. (1)

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

\begin{align*} \int {dy} &= \int {0\, dx} + c_3 \\ y &= c_3 \end{align*}

Figure 2.55: Slope field \(y^{\prime } = 0\)

Summary of solutions found

\begin{align*} y &= 0 \\ y &= c_3 \\ \end{align*}

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

ode:=y(x)*diff(y(x),x)^2 = 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 
trying 1st order linear 
<- 1st order linear successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int 0d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \end {array} \]

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

ode=y[x]*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.084 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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