Internal problem ID [2450]
Internal file name [OUTPUT/1942_Sunday_June_05_2022_02_40_14_AM_83224243/index.tex]
Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.1-2, page 6
Problem number: 1.1-2 (d).
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]
\[ \boxed {y^{\prime }=\sin \left (t \right )^{2}} \]
Integrating both sides gives \begin {align*} y &= \int { \sin \left (t \right )^{2}\,\mathop {\mathrm {d}t}}\\ &= \frac {t}{2}-\frac {\sin \left (2 t \right )}{4}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {t}{2}-\frac {\sin \left (2 t \right )}{4}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {t}{2}-\frac {\sin \left (2 t \right )}{4}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\sin \left (t \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime }d t =\int \sin \left (t \right )^{2}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\sin \left (t \right ) \cos \left (t \right )}{2}+\frac {t}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {\sin \left (t \right ) \cos \left (t \right )}{2}+\frac {t}{2}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 15
dsolve(diff(y(t),t)=sin(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {t}{2}+c_{1} -\frac {\sin \left (2 t \right )}{4} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 21
DSolve[y'[t]==Sin[t]^2,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {t}{2}-\frac {1}{4} \sin (2 t)+c_1 \]