Internal problem ID [2446]
Internal file name [OUTPUT/1938_Sunday_June_05_2022_02_40_07_AM_86851588/index.tex
]
Book: Elementary Differential Equations, Martin, Reissner, 2nd ed, 1961
Section: Exercis 2, page 5
Problem number: 3(i).
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]
\[ \boxed {y^{\prime } \sin \left (x \right )=1} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {1}{\sin \left (x \right )}\,\mathop {\mathrm {d}x}}\\ &= \ln \left (\tan \left (\frac {x}{2}\right )\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (\tan \left (\frac {x}{2}\right )\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \ln \left (\tan \left (\frac {x}{2}\right )\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sin \left (x \right )=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{\sin \left (x \right )} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{\sin \left (x \right )}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve(diff(y(x),x)*sin(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = -\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 13
DSolve[y'[x]*Sin[x]==1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\text {arctanh}(\cos (x))+c_1 \]