2.1.48 Problem 48

Internal problem ID [10306]
Book : First order enumerated odes
Section : section 1
Problem number : 48
Date solved : Monday, January 26, 2026 at 09:36:28 PM
CAS classification : [_quadrature]

2.1.48.1 Solved using first_order_ode_quadrature

0.091 (sec)

Entering first order ode quadrature solver

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

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_1 \\ y &= c_1 \end{align*}

Figure 2.39: Isoclines for \(x {y^{\prime }}^{n} = 0\)

Summary of solutions found

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

2.1.48.2 Solved using first_order_ode_homog_type_D2

0.398 (sec)

Entering first order ode homog type D2 solver

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

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

\begin{align*} x \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right )^{n} = 0 \end{align*}

Which is now solved The ode

\begin{equation} u^{\prime }\left (x \right ) = -\frac {u \left (x \right )}{x} \end{equation}

is separable as it can be written as

\begin{align*} u^{\prime }\left (x \right )&= -\frac {u \left (x \right )}{x}\\ &= f(x) g(u) \end{align*}

Where

\begin{align*} f(x) &= -\frac {1}{x}\\ g(u) &= u \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(u)} \,du} &= \int { f(x) \,dx} \\ \int { \frac {1}{u}\,du} &= \int { -\frac {1}{x} \,dx} \\ \end{align*}

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

Taking the exponential of both sides the solution becomes

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

We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is zero, since we had to divide by this above. Solving \(g(u)=0\) or

\[ u=0 \]

for \(u \left (x \right )\) gives

\begin{align*} u \left (x \right )&=0 \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

\begin{align*} u \left (x \right ) &= \frac {c_1}{x} \\ u \left (x \right ) &= 0 \\ \end{align*}

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

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

Converting \(u \left (x \right ) = 0\) back to \(y\) gives

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

Figure 2.40: Isoclines for \(x {y^{\prime }}^{n} = 0\)

Summary of solutions found

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

2.1.48.3 ✓ Maple. Time used: 0.002 (sec). Leaf size: 5

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

\[ y = c_1 \]

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (\frac {d}{d x}y \left (x \right )\right )^{10306}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int 0d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \end {array} \]

2.1.48.4 ✓ Mathematica. Time used: 0.003 (sec). Leaf size: 15

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

\begin{align*} y(x)&\to 0^{\frac {1}{n}} x+c_1 \end{align*}

2.1.48.5 ✗ Sympy

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)
 

Timed Out
 

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