2.14 problem 2(d)

Internal problem ID [6152]
Internal file name [OUTPUT/5400_Sunday_June_05_2022_03_36_18_PM_50028180/index.tex]

Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page 12
Problem number: 2(d).
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "exact", "separable", "differentialType", "first_order_ode_lie_symmetry_lookup"

Maple gives the following as the ode type

[_separable]

\[ \boxed {y^{\prime } y^{2}=x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}

2.14.1 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)\\ &= \frac {x +2}{y^{2}} \end {align*}

The \(x\) domain of \(f(x,y)\) when \(y=4\) is \[ \{-\infty

The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y=4\) is \[ \{-\infty

2.14.2 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x +2}{y^{2}} \end {align*}

Where \(f(x)=x +2\) and \(g(y)=\frac {1}{y^{2}}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y^{2}}} \,dy &= x +2 \,d x \\ \int { \frac {1}{\frac {1}{y^{2}}} \,dy} &= \int {x +2 \,d x} \\ \frac {y^{3}}{3}&=\frac {1}{2} x^{2}+2 x +c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\left (12 x^{2}+24 c_{1} +48 x \right )^{\frac {1}{3}}}{2} \\ y &= -\frac {\left (12 x^{2}+24 c_{1} +48 x \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (12 x^{2}+24 c_{1} +48 x \right )^{\frac {1}{3}}}{4} \\ y &= -\frac {\left (12 x^{2}+24 c_{1} +48 x \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (12 x^{2}+24 c_{1} +48 x \right )^{\frac {1}{3}}}{4} \\ \end{align*} Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=4\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 4 = -\frac {i \sqrt {3}\, c_{1}^{\frac {1}{3}} 24^{\frac {1}{3}}}{4}-\frac {c_{1}^{\frac {1}{3}} 24^{\frac {1}{3}}}{4} \end {align*}

Warning: Unable to solve for constant of integration. Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=4\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 4 = \frac {i \sqrt {3}\, c_{1}^{\frac {1}{3}} 24^{\frac {1}{3}}}{4}-\frac {c_{1}^{\frac {1}{3}} 24^{\frac {1}{3}}}{4} \end {align*}

Warning: Unable to solve for constant of integration. Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=4\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 4 = \frac {c_{1}^{\frac {1}{3}} 24^{\frac {1}{3}}}{2} \end {align*}

The solutions are \begin {align*} c_{1} = {\frac {64}{3}} \end {align*}

Trying the constant \begin {align*} c_{1} = {\frac {64}{3}} \end {align*}

Substituting this in the general solution gives \begin {align*} y&=\frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \end {align*}

The constant \(c_{1} = {\frac {64}{3}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \] Verified OK.

2.14.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime } y^{2}=x +2, y \left (0\right )=4\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } y^{2}d x =\int \left (x +2\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{3}}{3}=\frac {1}{2} x^{2}+2 x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (12 x^{2}+24 c_{1} +48 x \right )^{\frac {1}{3}}}{2} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=4 \\ {} & {} & 4=\frac {c_{1}^{\frac {1}{3}} 24^{\frac {1}{3}}}{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\frac {64}{3} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\frac {64}{3}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.141 (sec). Leaf size: 18

dsolve([y(x)^2*diff(y(x),x)=x+2,y(0) = 4],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \]

✓ Solution by Mathematica

Time used: 0.211 (sec). Leaf size: 21

DSolve[{y[x]^2*y'[x]==x+2,{y[0]==4}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt [3]{\frac {3 x^2}{2}+6 x+64} \]