1.18 problem 18

Internal problem ID [7062]
Internal file name [OUTPUT/6048_Sunday_June_05_2022_04_15_19_PM_15217227/index.tex]

Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 18.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y^{\prime }-\sqrt {\frac {y+1}{y^{2}}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}

1.18.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= \sqrt {\frac {1+y}{y^{2}}} \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=0\) is \[ \{-1\le y <0, 0

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=0\) is \[ \{-\infty \le y <-1, -1

1.18.2 Solving as quadrature ode

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

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=1\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} -\frac {2 \sqrt {2}}{3} = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = -\frac {2 \sqrt {2}}{3} \end {align*}

Trying the constant \begin {align*} c_{1} = -\frac {2 \sqrt {2}}{3} \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \frac {2 y \left (-2+y \right ) \sqrt {-\frac {-1-y}{y^{2}}}}{3} = x -\frac {2 \sqrt {2}}{3} \end {align*}

The constant \(c_{1} = -\frac {2 \sqrt {2}}{3}\) gives valid solution.

Summary

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

Verification of solutions

\[ \frac {2 y \left (-2+y\right ) \sqrt {\frac {y+1}{y^{2}}}}{3} = x -\frac {2 \sqrt {2}}{3} \] Verified OK.

1.18.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-\sqrt {\frac {y+1}{y^{2}}}=0, y \left (0\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sqrt {\frac {y+1}{y^{2}}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {\frac {y+1}{y^{2}}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {\frac {y+1}{y^{2}}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {2 \left (y+1\right ) \left (-2+y\right )}{3 \sqrt {\frac {y+1}{y^{2}}}\, y}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (-8+9 c_{1}^{2}+18 c_{1} x +9 x^{2}+3 \sqrt {9 c_{1}^{4}+36 c_{1}^{3} x +54 c_{1}^{2} x^{2}+36 c_{1} x^{3}+9 x^{4}-16 c_{1}^{2}-32 c_{1} x -16 x^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (-8+9 c_{1}^{2}+18 c_{1} x +9 x^{2}+3 \sqrt {9 c_{1}^{4}+36 c_{1}^{3} x +54 c_{1}^{2} x^{2}+36 c_{1} x^{3}+9 x^{4}-16 c_{1}^{2}-32 c_{1} x -16 x^{2}}\right )^{\frac {1}{3}}}+1 \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=\frac {\left (-8+9 c_{1}^{2}+3 \sqrt {9 c_{1}^{4}-16 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (-8+9 c_{1}^{2}+3 \sqrt {9 c_{1}^{4}-16 c_{1}^{2}}\right )^{\frac {1}{3}}}+1 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {9 \textit {\_Z}^{4}-16 \textit {\_Z}^{2}}\right )^{\frac {2}{3}}+4\right ) \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {9 \textit {\_Z}^{4}-16 \textit {\_Z}^{2}}\right )^{\frac {2}{3}}+4\right )\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {{\left (-8+9 {\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )}^{2}+18 \mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right ) x +9 x^{2}+3 \sqrt {9}\, \sqrt {\left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+\frac {4}{3}\right ) {\left (\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+x \right )}^{2} \left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )-\frac {4}{3}\right )}\right )}^{\frac {2}{3}}+2 {\left (-8+9 {\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )}^{2}+18 \mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right ) x +9 x^{2}+3 \sqrt {9}\, \sqrt {\left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+\frac {4}{3}\right ) {\left (\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+x \right )}^{2} \left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )-\frac {4}{3}\right )}\right )}^{\frac {1}{3}}+4}{2 {\left (-8+9 {\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )}^{2}+18 \mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right ) x +9 x^{2}+3 \sqrt {9}\, \sqrt {\left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+\frac {4}{3}\right ) {\left (\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+x \right )}^{2} \left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )-\frac {4}{3}\right )}\right )}^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {{\left (-8+9 {\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )}^{2}+18 \mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right ) x +9 x^{2}+3 \sqrt {9}\, \sqrt {\left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+\frac {4}{3}\right ) {\left (\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+x \right )}^{2} \left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )-\frac {4}{3}\right )}\right )}^{\frac {2}{3}}+2 {\left (-8+9 {\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )}^{2}+18 \mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right ) x +9 x^{2}+3 \sqrt {9}\, \sqrt {\left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+\frac {4}{3}\right ) {\left (\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+x \right )}^{2} \left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )-\frac {4}{3}\right )}\right )}^{\frac {1}{3}}+4}{2 {\left (-8+9 {\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )}^{2}+18 \mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right ) x +9 x^{2}+3 \sqrt {9}\, \sqrt {\left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+\frac {4}{3}\right ) {\left (\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )+x \right )}^{2} \left (x +\mathit {RootOf}\left (\left (-8+9 \textit {\_Z}^{2}+3 \sqrt {\textit {\_Z}^{2} \left (9 \textit {\_Z}^{2}-16\right )}\right )^{\frac {2}{3}}+4\right )-\frac {4}{3}\right )}\right )}^{\frac {1}{3}}} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.469 (sec). Leaf size: 148

dsolve([diff(y(x),x)=sqrt( (1+y(x))/y(x)^2),y(0) = 1],y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {\left (1+i \sqrt {3}\right ) \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{\frac {1}{3}}+4}{4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{\frac {1}{3}}} \]

✓ Solution by Mathematica

Time used: 0.097 (sec). Leaf size: 123

DSolve[{y'[x]==Sqrt[ (1+y[x])/y[x]^2],y[0]==1},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^2+\sqrt {81 x^4-216 \sqrt {2} x^3+288 x^2-64}-12 \sqrt {2} x}+\frac {i \left (\sqrt {3}+i\right )}{\sqrt [3]{9 x^2+\sqrt {81 x^4-216 \sqrt {2} x^3+288 x^2-64}-12 \sqrt {2} x}}+1 \]