1.16 problem 2(f)

Internal problem ID [3017]
Internal file name [OUTPUT/2509_Sunday_June_05_2022_03_17_18_AM_43139761/index.tex]

Book: Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section: Exercises, page 14
Problem number: 2(f).
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y^{\prime }={\mathrm e}^{x} \sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}

1.16.1 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) &=0\\ q(x) &={\mathrm e}^{x} \sin \left (x \right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime } = {\mathrm e}^{x} \sin \left (x \right ) \end {align*}

The domain of \(p(x)=0\) is \[ \{-\infty

1.16.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { {\mathrm e}^{x} \sin \left (x \right )\,\mathop {\mathrm {d}x}}\\ &= \frac {{\mathrm e}^{x} \left (-\cos \left (x \right )+\sin \left (x \right )\right )}{2}+c_{1} \end {align*}

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

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

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

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

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

Summary

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

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {{\mathrm e}^{x} \sin \left (x \right )}{2}-\frac {\cos \left (x \right ) {\mathrm e}^{x}}{2}+\frac {1}{2} \] Verified OK.

1.16.3 Maple step by step solution

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

Maple trace

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 17

dsolve([diff(y(x),x)=exp(x)*sin(x),y(0) = 0],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {1}{2}+\frac {{\mathrm e}^{x} \left (\sin \left (x \right )-\cos \left (x \right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 24

DSolve[{y'[x]==Exp[x]*Sin[x],y[0]==0},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (e^x \sin (x)-e^x \cos (x)+1\right ) \]