Internal problem ID [2434]
Internal file name [OUTPUT/1926_Sunday_June_05_2022_02_39_45_AM_20320997/index.tex]
Book: Elementary Differential Equations, Martin, Reissner, 2nd ed, 1961
Section: Exercis 2, page 5
Problem number: 2(c).
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]
\[ \boxed {y^{\prime }=\frac {2}{\sqrt {-x^{2}+1}}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {2}{\sqrt {-x^{2}+1}}\,\mathop {\mathrm {d}x}}\\ &= 2 \arcsin \left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \arcsin \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = 2 \arcsin \left (x \right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2}{\sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {2}{\sqrt {-x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=2 \arcsin \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \arcsin \left (x \right )+c_{1} \end {array} \]
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: 10
dsolve(diff(y(x),x)=2/sqrt(1-x^2),y(x), singsol=all)
\[ y \left (x \right ) = 2 \arcsin \left (x \right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 28
DSolve[y'[x]==2/Sqrt[1-x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -4 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )+c_1 \]