Internal problem ID [3210]
Internal file name [OUTPUT/2702_Sunday_June_05_2022_08_38_59_AM_39178359/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 66.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }-\left (\sin \left (x \right )^{2}-y\right ) \cos \left (x \right )=0} \]
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) &=\cos \left (x \right ) \sin \left (x \right )^{2} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+y \cos \left (x \right ) = \cos \left (x \right ) \sin \left (x \right )^{2} \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 (\cos \left (x \right ) \sin \left (x \right )^{2}\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 (\cos \left (x \right ) \sin \left (x \right )^{2}\right )\\ \mathrm {d} \left ({\mathrm e}^{\sin \left (x \right )} y\right ) &= \left (\cos \left (x \right ) \sin \left (x \right )^{2} {\mathrm e}^{\sin \left (x \right )}\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{\sin \left (x \right )} y &= \int {\cos \left (x \right ) \sin \left (x \right )^{2} {\mathrm e}^{\sin \left (x \right )}\,\mathrm {d} x}\\ {\mathrm e}^{\sin \left (x \right )} y &= \sin \left (x \right )^{2} {\mathrm e}^{\sin \left (x \right )}-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 (\sin \left (x \right )^{2} {\mathrm e}^{\sin \left (x \right )}-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 &= \sin \left (x \right )^{2}-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 &= \sin \left (x \right )^{2}-2 \sin \left (x \right )+2+c_{1} {\mathrm e}^{-\sin \left (x \right )} \\ \end{align*}
Verification of solutions
\[ y = \sin \left (x \right )^{2}-2 \sin \left (x \right )+2+c_{1} {\mathrm e}^{-\sin \left (x \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (\sin \left (x \right )^{2}-y\right ) \cos \left (x \right )=0 \\ \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 }=\left (\sin \left (x \right )^{2}-y\right ) \cos \left (x \right ) \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=-y \cos \left (x \right )+\cos \left (x \right ) \sin \left (x \right )^{2} \\ \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 )=\cos \left (x \right ) \sin \left (x \right )^{2} \\ \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 ) \cos \left (x \right ) \sin \left (x \right )^{2} \\ \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 ) \cos \left (x \right ) \sin \left (x \right )^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \mu \left (x \right ) \cos \left (x \right ) \sin \left (x \right )^{2}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) \cos \left (x \right ) \sin \left (x \right )^{2}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 \cos \left (x \right ) \sin \left (x \right )^{2} {\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 {\sin \left (x \right )^{2} {\mathrm e}^{\sin \left (x \right )}-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 )-\cos \left (x \right )^{2}+3+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: 23
dsolve(diff(y(x),x)=( sin(x)^2-y(x))*cos(x),y(x), singsol=all)
\[ y \left (x \right ) = 3+{\mathrm e}^{-\sin \left (x \right )} c_{1} -\cos \left (x \right )^{2}-2 \sin \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.147 (sec). Leaf size: 30
DSolve[y'[x]==( Sin[x]^2-y[x])*Cos[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -2 \sin (x)-\frac {1}{2} \cos (2 x)+c_1 e^{-\sin (x)}+\frac {5}{2} \]