2.15 problem 2(e)

Internal problem ID [6153]
Internal file name [OUTPUT/5401_Sunday_June_05_2022_03_36_20_PM_57043492/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(e).
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_separable]

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

2.15.1 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)\\ &= x^{2} y^{2} \end {align*}

The \(x\) domain of \(f(x,y)\) when \(y=2\) is \[ \{-\infty

The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y=2\) is \[ \{-\infty

2.15.2 Solving as separable ode

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

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

The solutions are \begin {align*} c_{1} = -{\frac {1}{6}} \end {align*}

Trying the constant \begin {align*} c_{1} = -{\frac {1}{6}} \end {align*}

Substituting this in the general solution gives \begin {align*} y&=-\frac {6}{2 x^{3}-1} \end {align*}

The constant \(c_{1} = -{\frac {1}{6}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {6}{2 x^{3}-1} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = -\frac {6}{2 x^{3}-1} \] Verified OK.

2.15.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-x^{2} y^{2}=0, y \left (-1\right )=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=x^{2} y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=x^{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=\frac {x^{3}}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {3}{x^{3}+3 c_{1}} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (-1\right )=2 \\ {} & {} & 2=-\frac {3}{3 c_{1} -1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =-\frac {1}{6} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =-\frac {1}{6}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {6}{2 x^{3}-1} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {6}{2 x^{3}-1} \end {array} \]

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: 0.031 (sec). Leaf size: 15

dsolve([diff(y(x),x)=x^2*y(x)^2,y(-1) = 2],y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {6}{2 x^{3}-1} \]

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 16

DSolve[{y'[x]==x^2*y[x]^2,{y[-1]==2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {6}{1-2 x^3} \]