Internal problem ID [6141]
Internal file name [OUTPUT/5389_Sunday_June_05_2022_03_36_01_PM_92610766/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(c).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }+y \tan \left (x \right )=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\tan \left (x \right ) y \end {align*}
Where \(f(x)=-\tan \left (x \right )\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\tan \left (x \right ) \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\tan \left (x \right ) \,d x}\\ \ln \left (y \right )&=\ln \left (\cos \left (x \right )\right )+c_{1}\\ y&={\mathrm e}^{\ln \left (\cos \left (x \right )\right )+c_{1}}\\ &=c_{1} \cos \left (x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \cos \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \cos \left (x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y \tan \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y \tan \left (x \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\tan \left (x \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\tan \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\ln \left (\cos \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{c_{1}} \cos \left (x \right ) \end {array} \]
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: 0.015 (sec). Leaf size: 8
dsolve(diff(y(x),x)+y(x)*tan(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \cos \left (x \right ) c_{1} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 15
DSolve[y'[x]+y[x]*Tan[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos (x) \\ y(x)\to 0 \\ \end{align*}