1.3 problem 3

Internal problem ID [5716]
Internal file name [OUTPUT/4964_Sunday_June_05_2022_03_15_22_PM_12531112/index.tex]

Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number: 3.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_separable]

\[ \boxed {y^{\prime }-y \sin \left (x \right )=0} \]

1.3.1 Solving as separable ode

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

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

Summary

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

Figure 3: Slope field plot

Verification of solutions

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

1.3.2 Maple step by step solution

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

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: 0.0 (sec). Leaf size: 11

dsolve(diff(y(x),x)=y(x)*sin(x),y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\cos \left (x \right )} \]

✓ Solution by Mathematica

Time used: 0.03 (sec). Leaf size: 19

DSolve[y'[x]==y[x]*Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 e^{-\cos (x)} \\ y(x)\to 0 \\ \end{align*}