2.1.3 Problem 3

Internal problem ID [10362]
Book : Second order enumerated odes
Section : section 1
Problem number : 3
Date solved : Monday, December 08, 2025 at 08:06:52 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )^{n}&=0 \\ \end{align*}

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} \frac {d^{2}}{d x^{2}}y \left (x \right )&=0 \\ \end{align*}

The above is now solved separately.

Solving Eq. (1)

Entering second order ode quadrature solverIntegrating twice gives the solution

\[ y \left (x \right )= \textit {\_C16} x + \textit {\_C17} \]

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

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

\[ y = c_1 x +c_2 \]

Maple trace

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

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )^{10362}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {0\pm \left (\sqrt {0}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =0 \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )=1 \\ \bullet & {} & \textrm {Repeated root, multiply}\hspace {3pt} y_{1}\left (x \right )\hspace {3pt}\textrm {by}\hspace {3pt} x \hspace {3pt}\textrm {to ensure linear independence}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )=x \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} y_{1}\left (x \right )+\mathit {C2} y_{2}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C2} x +\mathit {C1} \end {array} \]

2.1.3.2 ✓ Mathematica. Time used: 0.003 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to \frac {1}{2} 0^{\frac {1}{n}} x^2+c_2 x+c_1 \end{align*}

2.1.3.3 ✗ Sympy

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

TypeError : cannot determine truth value of Relational: n > 1