Internal problem ID [14082]
Internal file name [OUTPUT/13763_Saturday_March_02_2024_02_49_19_PM_22773647/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 46.
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]
\[ \boxed {y^{\prime }=\sin \left (x \right )^{3} \tan \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \sin \left (x \right )^{3} \tan \left (x \right )\,\mathop {\mathrm {d}x}}\\ &= -\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\sin \left (x \right )^{3} \tan \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sin \left (x \right )^{3} \tan \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \end {array} \]
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: 22
dsolve(diff(y(x),x)=sin(x)^3*tan(x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 23
DSolve[y'[x]==Sin[x]^3*Tan[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {arctanh}(\sin (x))-\frac {1}{3} \sin ^3(x)-\sin (x)+c_1 \]