Internal problem ID [6146]
Internal file name [OUTPUT/5394_Sunday_June_05_2022_03_36_09_PM_7071013/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(h).
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}}\\ &=\frac {c_{1}}{\cos \left (x \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{\cos \left (x \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{\cos \left (x \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \tan \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \tan \left (x \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\tan \left (x \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \tan \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (\cos \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{\cos \left (x \right )} \end {array} \]
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: 0.016 (sec). Leaf size: 8
dsolve(diff(y(x),x)-y(x)*tan(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sec \left (x \right ) c_{1} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 15
DSolve[y'[x]-y[x]*Tan[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \sec (x) \\ y(x)\to 0 \\ \end{align*}