Internal problem ID [13442]
Internal file name [OUTPUT/12614_Friday_February_16_2024_12_00_17_AM_73858456/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 8. Review exercises for part of part II. page 143
Problem number: 21.
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]
\[ \boxed {2 \cos \left (x \right ) y^{\prime }=-\sin \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { -\frac {\sin \left (x \right )}{2 \cos \left (x \right )}\,\mathop {\mathrm {d}x}}\\ &= \frac {\ln \left (\cos \left (x \right )\right )}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\ln \left (\cos \left (x \right )\right )}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\ln \left (\cos \left (x \right )\right )}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 \cos \left (x \right ) y^{\prime }=-\sin \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\sin \left (x \right )}{2 \cos \left (x \right )} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {\sin \left (x \right )}{2 \cos \left (x \right )}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\ln \left (\cos \left (x \right )\right )}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\ln \left (\cos \left (x \right )\right )}{2}+c_{1} \end {array} \]
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: 11
dsolve(sin(x)+2*cos(x)*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\ln \left (\cos \left (x \right )\right )}{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 15
DSolve[Sin[x]+2*Cos[x]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \log (\cos (x))+c_1 \]