Internal problem ID [6145]
Internal file name [OUTPUT/5393_Sunday_June_05_2022_03_36_07_PM_45941527/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(g).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime } \sin \left (y\right )=x^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x^{2}}{\sin \left (y \right )} \end {align*}
Where \(f(x)=x^{2}\) and \(g(y)=\frac {1}{\sin \left (y \right )}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{\sin \left (y \right )}} \,dy &= x^{2} \,d x \\ \int { \frac {1}{\frac {1}{\sin \left (y \right )}} \,dy} &= \int {x^{2} \,d x} \\ -\cos \left (y \right )&=\frac {x^{3}}{3}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \pi -\arccos \left (\frac {x^{3}}{3}+c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \pi -\arccos \left (\frac {x^{3}}{3}+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \pi -\arccos \left (\frac {x^{3}}{3}+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sin \left (y\right )=x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } \sin \left (y\right )d x =\int x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\cos \left (y\right )=\frac {x^{3}}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\pi -\arccos \left (\frac {x^{3}}{3}+c_{1} \right ) \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 16
dsolve(diff(y(x),x)*sin(y(x))=x^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\pi }{2}+\arcsin \left (\frac {x^{3}}{3}+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.499 (sec). Leaf size: 37
DSolve[y'[x]*Sin[y[x]]==x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\arccos \left (-\frac {x^3}{3}-c_1\right ) \\ y(x)\to \arccos \left (-\frac {x^3}{3}-c_1\right ) \\ \end{align*}