Internal problem ID [13272]
Internal file name [OUTPUT/12444_Wednesday_February_14_2024_02_06_17_AM_21519458/index.tex
]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page
32
Problem number: 2.5 (b 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]
\[ \boxed {y^{\prime }=\sin \left (\frac {x}{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=0\\ q(x) &=\sin \left (\frac {x}{2}\right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime } = \sin \left (\frac {x}{2}\right ) \end {align*}
The domain of \(p(x)=0\) is \[
\{-\infty
Integrating both sides gives \begin {align*} y &= \int { \sin \left (\frac {x}{2}\right )\,\mathop {\mathrm {d}x}}\\ &= -2 \cos \left (\frac {x}{2}\right )+c_{1} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 0 = c_{1} -2 \end {align*}
The solutions are \begin {align*} c_{1} = 2 \end {align*}
Trying the constant \begin {align*} c_{1} = 2 \end {align*}
Substituting this in the general solution gives \begin {align*} y&=-2 \cos \left (\frac {x}{2}\right )+2 \end {align*}
The constant \(c_{1} = 2\) gives valid solution.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= -2 \cos \left (\frac {x}{2}\right )+2 \\
\end{align*} Verification of solutions
\[
y = -2 \cos \left (\frac {x}{2}\right )+2
\] Verified OK. \[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\sin \left (\frac {x}{2}\right ), y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sin \left (\frac {x}{2}\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-2 \cos \left (\frac {x}{2}\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-2 \cos \left (\frac {x}{2}\right )+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} -2 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =2 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =2\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-2 \cos \left (\frac {x}{2}\right )+2 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-2 \cos \left (\frac {x}{2}\right )+2 \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 12
\[
y \left (x \right ) = -2 \cos \left (\frac {x}{2}\right )+2
\]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 15
\[
y(x)\to 4 \sin ^2\left (\frac {x}{4}\right )
\]
1.31.2 Solving as quadrature ode
1.31.3 Maple step by step solution
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
<- quadrature successful`
dsolve([diff(y(x),x)=sin(x/2),y(0) = 0],y(x), singsol=all)
DSolve[{y'[x]==Sin[x/2],{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]