Internal problem ID [13271]
Internal file name [OUTPUT/12443_Wednesday_February_14_2024_02_06_17_AM_83170867/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 (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]
\[ \boxed {y^{\prime }=\sin \left (\frac {x}{2}\right )} \]
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*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -2 \cos \left (\frac {x}{2}\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -2 \cos \left (\frac {x}{2}\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\sin \left (\frac {x}{2}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sin \left (\frac {x}{2}\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-2 \cos \left (\frac {x}{2}\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-2 \cos \left (\frac {x}{2}\right )+c_{1} \end {array} \]
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: 12
dsolve(diff(y(x),x)=sin(x/2),y(x), singsol=all)
\[ y \left (x \right ) = -2 \cos \left (\frac {x}{2}\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 16
DSolve[y'[x]==Sin[x/2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -2 \cos \left (\frac {x}{2}\right )+c_1 \]