Internal problem ID [7128]
Internal file name [OUTPUT/6114_Sunday_June_05_2022_04_23_18_PM_8884940/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 83.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }-\sqrt {1-y^{2}}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \sqrt {-y^{2}+1} \end {align*}
Where \(f(x)=1\) and \(g(y)=\sqrt {-y^{2}+1}\). Integrating both sides gives \begin{align*} \frac {1}{\sqrt {-y^{2}+1}} \,dy &= 1 \,d x \\ \int { \frac {1}{\sqrt {-y^{2}+1}} \,dy} &= \int {1 \,d x} \\ \arcsin \left (y \right )&=x +c_{1} \\ \end{align*} Which results in \begin{align*} y &= \sin \left (x +c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sin \left (x +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \sin \left (x +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {1-y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sqrt {1-y^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {1-y^{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {1-y^{2}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \arcsin \left (y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sin \left (x +c_{1} \right ) \end {array} \]
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: 8
dsolve(diff(y(x),x)=sqrt(1-y(x)^2),y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.228 (sec). Leaf size: 28
DSolve[y'[x]==Sqrt[1-y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \cos (x+c_1) \\ y(x)\to -1 \\ y(x)\to 1 \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}