2.1.47 Problem 47

Internal problem ID [9031]
Book : First order enumerated odes
Section : section 1
Problem number : 47
Date solved : Sunday, March 30, 2025 at 01:59:39 PM
CAS classification : [_quadrature]

Solve

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

Unable to solve. Terminating.

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

ode:=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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 15

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

$$y(x)\to 0^{\frac{1}{n}}x+c_1$$

✗ Sympy

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

TypeError : cannot determine truth value of Relational