Internal problem ID [924]
Internal file name [OUTPUT/924_Sunday_June_05_2022_01_54_07_AM_72624317/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number: 48(a).
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]
\[ \boxed {\sec \left (y\right )^{2} y^{\prime }-3 \tan \left (y\right )=-1} \]
Integrating both sides gives \begin {align*} \int \frac {\sec \left (y \right )^{2}}{3 \tan \left (y \right )-1}d y &= x +c_{1}\\ \frac {\ln \left (3 \tan \left (y \right )-1\right )}{3}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\arctan \left (\frac {{\mathrm e}^{3 x +3 c_{1}}}{3}+\frac {1}{3}\right )\\ &=\arctan \left (\frac {{\mathrm e}^{3 x} c_{1}^{3}}{3}+\frac {1}{3}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \arctan \left (\frac {{\mathrm e}^{3 x} c_{1}^{3}}{3}+\frac {1}{3}\right ) \\ \end{align*}
Verification of solutions
\[ y = \arctan \left (\frac {{\mathrm e}^{3 x} c_{1}^{3}}{3}+\frac {1}{3}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sec \left (y\right )^{2} y^{\prime }-3 \tan \left (y\right )=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 \tan \left (y\right )-1}{\sec \left (y\right )^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \sec \left (y\right )^{2}}{3 \tan \left (y\right )-1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \sec \left (y\right )^{2}}{3 \tan \left (y\right )-1}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (3 \tan \left (y\right )-1\right )}{3}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arctan \left (\frac {{\mathrm e}^{3 x +3 c_{1}}}{3}+\frac {1}{3}\right ) \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve(sec(y(x))^2*diff(y(x),x)-3*tan(y(x))= -1,y(x), singsol=all)
\[ y \left (x \right ) = \arctan \left (\frac {c_{1} {\mathrm e}^{3 x}}{3}+\frac {1}{3}\right ) \]
✓ Solution by Mathematica
Time used: 60.217 (sec). Leaf size: 177
DSolve[Sec[y[x]]^2*y'[x]-3*Tan[y[x]]== -1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\arccos \left (-\frac {3 e^{6 c_1}}{\sqrt {e^{6 x}-2 e^{3 x+6 c_1}+10 e^{12 c_1}}}\right ) \\ y(x)\to \arccos \left (-\frac {3 e^{6 c_1}}{\sqrt {e^{6 x}-2 e^{3 x+6 c_1}+10 e^{12 c_1}}}\right ) \\ y(x)\to -\arccos \left (\frac {3 e^{6 c_1}}{\sqrt {e^{6 x}-2 e^{3 x+6 c_1}+10 e^{12 c_1}}}\right ) \\ y(x)\to \arccos \left (\frac {3 e^{6 c_1}}{\sqrt {e^{6 x}-2 e^{3 x+6 c_1}+10 e^{12 c_1}}}\right ) \\ \end{align*}