2.2 problem 2

Internal problem ID [5088]
Internal file name [OUTPUT/4581_Sunday_June_05_2022_03_01_17_PM_19661026/index.tex]

Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section: Program 24. First order differential equations. Further problems 24. page 1068
Problem number: 2.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "exact", "linear", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"

Maple gives the following as the ode type

[_separable]

\[ \boxed {\left (x^{3}+1\right ) y^{\prime }-x^{2} y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}

2.2.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}

Where here \begin {align*} p(x) &=-\frac {x^{2}}{x^{3}+1}\\ q(x) &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime }-\frac {x^{2} y}{x^{3}+1} = 0 \end {align*}

The domain of \(p(x)=-\frac {x^{2}}{x^{3}+1}\) is \[ \{x <-1\boldsymbol {\lor }-1

2.2.2 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x^{2} y}{x^{3}+1} \end {align*}

Where \(f(x)=\frac {x^{2}}{x^{3}+1}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {x^{2}}{x^{3}+1} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {x^{2}}{x^{3}+1} \,d x}\\ \ln \left (y \right )&=\frac {\ln \left (x^{3}+1\right )}{3}+c_{1}\\ y&={\mathrm e}^{\frac {\ln \left (x^{3}+1\right )}{3}+c_{1}}\\ &=c_{1} \left (x^{3}+1\right )^{\frac {1}{3}} \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 = 2^{\frac {1}{3}} c_{1} \end {align*}

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

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

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

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

Summary

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

(a) Solution plot

(b) Slope field plot

Verification of solutions

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

2.2.3 Maple step by step solution

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

Maple trace

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 15

dsolve([(1+x^3)*diff(y(x),x)=x^2*y(x),y(1) = 2],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 20

DSolve[{(1+x^3)*y'[x]==x^2*y[x],{y[1]==2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2^{2/3} \sqrt [3]{x^3+1} \]