Internal problem ID [4771]
Internal file name [OUTPUT/4264_Sunday_June_05_2022_12_49_53_PM_71133227/index.tex]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 3. Linear First-Order Equations. page
403
Problem number: 11.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }+y \cos \left (x \right )=\sin \left (2 x \right )} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=\cos \left (x \right )\\ q(x) &=\sin \left (2 x \right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime }+y \cos \left (x \right ) = \sin \left (2 x \right ) \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \cos \left (x \right )d x} \\ &= {\mathrm e}^{\sin \left (x \right )} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\sin \left (2 x \right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left ({\mathrm e}^{\sin \left (x \right )} y\right ) &= \left ({\mathrm e}^{\sin \left (x \right )}\right ) \left (\sin \left (2 x \right )\right )\\ \mathrm {d} \left ({\mathrm e}^{\sin \left (x \right )} y\right ) &= \left (\sin \left (2 x \right ) {\mathrm e}^{\sin \left (x \right )}\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{\sin \left (x \right )} y &= \int {\sin \left (2 x \right ) {\mathrm e}^{\sin \left (x \right )}\,\mathrm {d} x}\\ {\mathrm e}^{\sin \left (x \right )} y &= 2 \sin \left (x \right ) {\mathrm e}^{\sin \left (x \right )}-2 \,{\mathrm e}^{\sin \left (x \right )} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\sin \left (x \right )}\) results in \begin {align*} y &= {\mathrm e}^{-\sin \left (x \right )} \left (2 \sin \left (x \right ) {\mathrm e}^{\sin \left (x \right )}-2 \,{\mathrm e}^{\sin \left (x \right )}\right )+c_{1} {\mathrm e}^{-\sin \left (x \right )} \end {align*}
which simplifies to \begin {align*} y &= 2 \sin \left (x \right )-2+c_{1} {\mathrm e}^{-\sin \left (x \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \sin \left (x \right )-2+c_{1} {\mathrm e}^{-\sin \left (x \right )} \\ \end{align*}
Verification of solutions
\[ y = 2 \sin \left (x \right )-2+c_{1} {\mathrm e}^{-\sin \left (x \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y \cos \left (x \right )=\sin \left (2 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 }=-y \cos \left (x \right )+\sin \left (2 x \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }+y \cos \left (x \right )=\sin \left (2 x \right ) \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+y \cos \left (x \right )\right )=\mu \left (x \right ) \sin \left (2 x \right ) \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+y \cos \left (x \right )\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=\mu \left (x \right ) \cos \left (x \right ) \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )={\mathrm e}^{\sin \left (x \right )} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int \mu \left (x \right ) \sin \left (2 x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \mu \left (x \right ) \sin \left (2 x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) \sin \left (2 x \right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )={\mathrm e}^{\sin \left (x \right )} \\ {} & {} & y=\frac {\int \sin \left (2 x \right ) {\mathrm e}^{\sin \left (x \right )}d x +c_{1}}{{\mathrm e}^{\sin \left (x \right )}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {2 \sin \left (x \right ) {\mathrm e}^{\sin \left (x \right )}-2 \,{\mathrm e}^{\sin \left (x \right )}+c_{1}}{{\mathrm e}^{\sin \left (x \right )}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=2 \sin \left (x \right )-2+c_{1} {\mathrm e}^{-\sin \left (x \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve(diff(y(x),x)+y(x)*cos(x)=sin(2*x),y(x), singsol=all)
\[ y \left (x \right ) = 2 \sin \left (x \right )-2+{\mathrm e}^{-\sin \left (x \right )} c_{1} \]
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 20
DSolve[y'[x]+y[x]*Cos[x]==Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2 \sin (x)+c_1 e^{-\sin (x)}-2 \]