2.1.8 problem 8

Internal problem ID [8491]
Book : Second order enumerated odes
Section : section 1
Problem number : 8
Date solved : Sunday, November 10, 2024 at 03:55:02 AM
CAS classification : [[_2nd_order, _quadrature]]

Solve

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

Solved as second order missing y ode

Time used: 0.184 (sec)

This is second order ode with missing dependent variable \(y\). Let

\begin{align*} p(x) &= y^{\prime } \end{align*}

Then

\begin{align*} p'(x) &= y^{\prime \prime } \end{align*}

Hence the ode becomes

\begin{align*} {p^{\prime }\left (x \right )}^{2}-1 = 0 \end{align*}

Which is now solve for \(p(x)\) as first order ode. Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} p^{\prime }\left (x \right )&=1 \\ \tag{2} p^{\prime }\left (x \right )&=-1 \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

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

\begin{align*} \int {dp} &= \int {1\, dx}\\ p \left (x \right ) &= x + c_1 \end{align*}

Solving Eq. (2)

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

\begin{align*} \int {dp} &= \int {-1\, dx}\\ p \left (x \right ) &= -x + c_2 \end{align*}

For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = -x +c_2 \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 {-x +c_2\, dx}\\ y &= -\frac {1}{2} x^{2}+c_2 x + c_3 \end{align*}

For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} y^{\prime } = x +c_1 \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 {x +c_1\, dx}\\ y &= \frac {1}{2} x^{2}+c_1 x + c_4 \end{align*}

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= -\frac {1}{2} x^{2}+c_2 x +c_3 \\ y &= \frac {1}{2} x^{2}+c_1 x +c_4 \\ \end{align*}

Maple step by step solution

Maple trace

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

Maple dsolve solution

Solving time : 0.005 (sec)
Leaf size : 27

dsolve(diff(diff(y(x),x),x)^2 = 1, 
       y(x),singsol=all)
 

\begin{align*} y &= \frac {1}{2} x^{2}+c_{1} x +c_{2} \\ y &= -\frac {1}{2} x^{2}+c_{1} x +c_{2} \\ \end{align*}

Mathematica DSolve solution

Solving time : 0.003 (sec)
Leaf size : 37

DSolve[{(D[y[x],{x,2}])^2==1,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {x^2}{2}+c_2 x+c_1 \\ y(x)\to \frac {x^2}{2}+c_2 x+c_1 \\ \end{align*}