1.10 problem 10

Internal problem ID [7399]
Internal file name [OUTPUT/6366_Sunday_June_05_2022_04_41_48_PM_18474351/index.tex]

Book: Second order enumerated odes
Section: section 1
Problem number: 10.
ODE order: 2.
ODE degree: 2.

The type(s) of ODE detected by this program : "second_order_ode_high_degree", "second_order_ode_missing_y"

Maple gives the following as the ode type

[[_2nd_order, _quadrature]]

\[ \boxed {{y^{\prime \prime }}^{2}=x} \]

1.10.1 Solving as second order ode missing y ode

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}-x = 0 \end {align*}

Which is now solve for \(p(x)\) as first order ode. Solving the given ode for \(p^{\prime }\left (x \right )\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} p^{\prime }\left (x \right )&=\sqrt {x} \tag {1} \\ p^{\prime }\left (x \right )&=-\sqrt {x} \tag {2} \end {align*}

Now each one of the above ODE is solved.

Solving equation (1)

Integrating both sides gives \begin {align*} p \left (x \right ) &= \int { \sqrt {x}\,\mathop {\mathrm {d}x}}\\ &= \frac {2 x^{\frac {3}{2}}}{3}+c_{1} \end {align*}

Solving equation (2)

Integrating both sides gives \begin {align*} p \left (x \right ) &= \int { -\sqrt {x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {2 x^{\frac {3}{2}}}{3}+c_{2} \end {align*}

For solution (1) found earlier, since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = \frac {2 x^{\frac {3}{2}}}{3}+c_{1} \end {align*}

Integrating both sides gives \begin {align*} y &= \int { \frac {2 x^{\frac {3}{2}}}{3}+c_{1}\,\mathop {\mathrm {d}x}}\\ &= \frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{3} \end {align*}

Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = -\frac {2 x^{\frac {3}{2}}}{3}+c_{2} \end {align*}

Integrating both sides gives \begin {align*} y &= \int { -\frac {2 x^{\frac {3}{2}}}{3}+c_{2}\,\mathop {\mathrm {d}x}}\\ &= -\frac {4 x^{\frac {5}{2}}}{15}+c_{2} x +c_{4} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{3} \\ \tag{2} y &= -\frac {4 x^{\frac {5}{2}}}{15}+c_{2} x +c_{4} \\ \end{align*}

Verification of solutions

\[ y = \frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{3} \] Verified OK.

\[ y = -\frac {4 x^{\frac {5}{2}}}{15}+c_{2} x +c_{4} \] Verified OK.

1.10.2 Solving using Kovacic algorithm

Solving for \(y^{\prime \prime }\) from the ode gives \begin{align*} \tag{1} y^{\prime \prime } &= \sqrt {x} \\ \tag{2} y^{\prime \prime } &= -\sqrt {x} \\ \end{align*} Now each ode is solved. Integrating once gives \[ y^{\prime }= \frac {2 x^{\frac {3}{2}}}{3} + c_{1} \] Integrating again gives \[ y= \frac {4 x^{\frac {5}{2}}}{15} + c_{1} x + c_{2} \] Integrating once gives \[ y^{\prime }= -\frac {2 x^{\frac {3}{2}}}{3} + c_{3} \] Integrating again gives \[ y= -\frac {4 x^{\frac {5}{2}}}{15} + c_{3} x + c_{4} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{2} \\ \tag{2} y &= -\frac {4 x^{\frac {5}{2}}}{15}+c_{3} x +c_{4} \\ \end{align*}

Verification of solutions

\[ y = \frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{2} \] Verified OK.

\[ y = -\frac {4 x^{\frac {5}{2}}}{15}+c_{3} x +c_{4} \] Verified OK.

Maple trace

`Methods for second order ODEs: 
Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. 
   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   <- quadrature successful 
------------------- 
* Tackling next ODE. 
   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   <- quadrature successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 27

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

\begin{align*} y \left (x \right ) &= \frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{2} \\ y \left (x \right ) &= -\frac {4 x^{\frac {5}{2}}}{15}+c_{1} x +c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 41

DSolve[(y''[x])^2==x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {4 x^{5/2}}{15}+c_2 x+c_1 \\ y(x)\to \frac {4 x^{5/2}}{15}+c_2 x+c_1 \\ \end{align*}