Internal problem ID [875]
Internal file name [OUTPUT/875_Sunday_June_05_2022_01_52_59_AM_1317647/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 1, Introduction. Section 1.2 Page 14
Problem number: 3(b).
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]
\[ \boxed {y^{\prime }=-\sin \left (x \right ) x} \]
Integrating both sides gives \begin {align*} y &= \int { -\sin \left (x \right ) x\,\mathop {\mathrm {d}x}}\\ &= x \cos \left (x \right )-\sin \left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x \cos \left (x \right )-\sin \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = x \cos \left (x \right )-\sin \left (x \right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=-\sin \left (x \right ) x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\sin \left (x \right ) x d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x \cos \left (x \right )-\sin \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x \cos \left (x \right )-\sin \left (x \right )+c_{1} \end {array} \]
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: 14
dsolve(diff(y(x),x) = -x*sin(x),y(x), singsol=all)
\[ y \left (x \right ) = -\sin \left (x \right )+\cos \left (x \right ) x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 16
DSolve[y'[x] == -x*Sin[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\sin (x)+x \cos (x)+c_1 \]