2.13 problem 13

Internal problem ID [5099]
Internal file name [OUTPUT/4592_Sunday_June_05_2022_03_01_28_PM_86121755/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: 13.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_linear]

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

2.13.1 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 {1}{x}\\ q(x) &=x^{2} \cos \left (x \right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime }-\frac {y}{x} = x^{2} \cos \left (x \right ) \end {align*}

The domain of \(p(x)=-\frac {1}{x}\) is \[ \{x <0\boldsymbol {\lor }0

2.13.2 Solving as linear ode

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

Integrating gives \begin {align*} \frac {y}{x} &= \int {\cos \left (x \right ) x\,\mathrm {d} x}\\ \frac {y}{x} &= \cos \left (x \right )+\sin \left (x \right ) x + c_{1} \end {align*}

Dividing both sides by the integrating factor \(\mu =\frac {1}{x}\) results in \begin {align*} y &= x \left (\cos \left (x \right )+\sin \left (x \right ) x \right )+c_{1} x \end {align*}

which simplifies to \begin {align*} y &= x \left (\cos \left (x \right )+\sin \left (x \right ) x +c_{1} \right ) \end {align*}

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

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

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

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

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

Summary

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

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = x \left (\cos \left (x \right )+\sin \left (x \right ) x +1\right ) \] Verified OK.

2.13.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x y^{\prime }-y=x^{3} \cos \left (x \right ), y \left (\pi \right )=0\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 {y+x^{3} \cos \left (x \right )}{x} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y}{x}+x^{2} \cos \left (x \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {y}{x}=x^{2} \cos \left (x \right ) \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }-\frac {y}{x}\right )=\mu \left (x \right ) x^{2} \cos \left (x \right ) \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }-\frac {y}{x}\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=-\frac {\mu \left (x \right )}{x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int \mu \left (x \right ) x^{2} \cos \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \mu \left (x \right ) x^{2} \cos \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) x^{2} \cos \left (x \right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\frac {1}{x} \\ {} & {} & y=x \left (\int \cos \left (x \right ) x d x +c_{1} \right ) \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=x \left (\cos \left (x \right )+\sin \left (x \right ) x +c_{1} \right ) \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (\pi \right )=0 \\ {} & {} & 0=\pi \left (-1+c_{1} \right ) \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =1 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =1\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=x \left (\cos \left (x \right )+\sin \left (x \right ) x +1\right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=x \left (\cos \left (x \right )+\sin \left (x \right ) x +1\right ) \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: 0.016 (sec). Leaf size: 14

dsolve([x*diff(y(x),x)-y(x)=x^3*cos(x),y(Pi) = 0],y(x), singsol=all)
 

\[ y \left (x \right ) = \left (\cos \left (x \right )+\sin \left (x \right ) x +1\right ) x \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 15

DSolve[{x*y'[x]-y[x]==x^3*Cos[x],{y[Pi]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x (x \sin (x)+\cos (x)+1) \]