2.14 problem 14

Internal problem ID [5100]
Internal file name [OUTPUT/4593_Sunday_June_05_2022_03_01_29_PM_28386368/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: 14.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_separable]

\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+3 x y=5 x} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}

2.14.1 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 {3 x}{x^{2}+1}\\ q(x) &=\frac {5 x}{x^{2}+1} \end {align*}

Hence the ode is \begin {align*} y^{\prime }+\frac {3 x y}{x^{2}+1} = \frac {5 x}{x^{2}+1} \end {align*}

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

2.14.2 Solving as linear ode

Entering Linear first order ODE solver. The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {3 x}{x^{2}+1}d x} \\ &= \left (x^{2}+1\right )^{\frac {3}{2}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {5 x}{x^{2}+1}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\left (x^{2}+1\right )^{\frac {3}{2}} y\right ) &= \left (\left (x^{2}+1\right )^{\frac {3}{2}}\right ) \left (\frac {5 x}{x^{2}+1}\right )\\ \mathrm {d} \left (\left (x^{2}+1\right )^{\frac {3}{2}} y\right ) &= \left (5 \sqrt {x^{2}+1}\, x\right )\, \mathrm {d} x \end {align*}

Integrating gives \begin {align*} \left (x^{2}+1\right )^{\frac {3}{2}} y &= \int {5 \sqrt {x^{2}+1}\, x\,\mathrm {d} x}\\ \left (x^{2}+1\right )^{\frac {3}{2}} y &= \frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}}{3} + c_{1} \end {align*}

Dividing both sides by the integrating factor \(\mu =\left (x^{2}+1\right )^{\frac {3}{2}}\) results in \begin {align*} y &= \frac {5}{3}+\frac {c_{1}}{\left (x^{2}+1\right )^{\frac {3}{2}}} \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 {5}{3}+\frac {\sqrt {2}\, c_{1}}{4} \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 this in the general solution gives \begin {align*} y&=\frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}+2 \sqrt {2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}}} \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} y &= \frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}+2 \sqrt {2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}}} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}+2 \sqrt {2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}}} \] Verified OK.

2.14.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\left (x^{2}+1\right ) y^{\prime }+3 x y=5 x , y \left (1\right )=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3 x y+5 x}{x^{2}+1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{3 y-5}=-\frac {x}{x^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{3 y-5}d x =\int -\frac {x}{x^{2}+1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (3 y-5\right )}{3}=-\frac {\ln \left (x^{2}+1\right )}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{-\frac {3 \ln \left (x^{2}+1\right )}{2}+3 c_{1}}}{3}+\frac {5}{3} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=2 \\ {} & {} & 2=\frac {{\mathrm e}^{-\frac {3 \ln \left (2\right )}{2}+3 c_{1}}}{3}+\frac {5}{3} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\frac {\ln \left (2\right )}{2} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\frac {\ln \left (2\right )}{2}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {5}{3}+\frac {2 \sqrt {2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}}} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {5}{3}+\frac {2 \sqrt {2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}}} \end {array} \]

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: 0.031 (sec). Leaf size: 18

dsolve([(1+x^2)*diff(y(x),x)+3*x*y(x)=5*x,y(1) = 2],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {5}{3}+\frac {2 \sqrt {2}}{3 \left (x^{2}+1\right )^{\frac {3}{2}}} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 27

DSolve[{(1+x^2)*y'[x]+3*x*y[x]==5*x,{y[1]==2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {2 \sqrt {2}}{3 \left (x^2+1\right )^{3/2}}+\frac {5}{3} \]