Internal problem ID [7354]
Internal file name [OUTPUT/6335_Sunday_June_05_2022_04_40_29_PM_52270518/index.tex]
Book: First order enumerated odes
Section: section 1
Problem number: 38.
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]
\[ \boxed {x y^{\prime }=\sin \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {\sin \left (x \right )}{x}\,\mathop {\mathrm {d}x}}\\ &= \operatorname {Si}\left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \operatorname {Si}\left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \operatorname {Si}\left (x \right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }=\sin \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sin \left (x \right )}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sin \left (x \right )}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\mathrm {Si}\left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\mathrm {Si}\left (x \right )+c_{1} \end {array} \]
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: 8
dsolve(x*diff(y(x),x)=sin(x),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {Si}\left (x \right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 10
DSolve[x*y'[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {Si}(x)+c_1 \]