Internal problem ID [9]
Internal file name [OUTPUT/9_Sunday_June_05_2022_01_33_42_AM_384554/index.tex
]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.2. Integrals as general and particular solutions. Page 16
Problem number: 9.
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]
\[ \boxed {y^{\prime }=\frac {1}{\sqrt {-x^{2}+1}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=0\\ q(x) &=\frac {1}{\sqrt {-x^{2}+1}} \end {align*}
Hence the ode is \begin {align*} y^{\prime } = \frac {1}{\sqrt {-x^{2}+1}} \end {align*}
The domain of \(p(x)=0\) is \[
\{-\infty
Integrating both sides gives \begin {align*} y &= \int { \frac {1}{\sqrt {-x^{2}+1}}\,\mathop {\mathrm {d}x}}\\ &= \arcsin \left (x \right )+c_{1} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 0 = c_{1} \end {align*}
The solutions are \begin {align*} c_{1} = 0 \end {align*}
Trying the constant \begin {align*} c_{1} = 0 \end {align*}
Substituting this in the general solution gives \begin {align*} y&=\arcsin \left (x \right ) \end {align*}
The constant \(c_{1} = 0\) gives valid solution.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= \arcsin \left (x \right ) \\
\end{align*} Verification of solutions
\[
y = \arcsin \left (x \right )
\] Verified OK. \[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {1}{\sqrt {-x^{2}+1}}, y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{\sqrt {-x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\arcsin \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arcsin \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\arcsin \left (x \right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\arcsin \left (x \right ) \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 6
\[
y \left (x \right ) = \arcsin \left (x \right )
\]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 31
\[
y(x)\to \frac {1}{2} \left (\pi -4 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )\right )
\]
1.9.2 Solving as quadrature ode
1.9.3 Maple step by step solution
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
<- quadrature successful`
dsolve([diff(y(x),x) = 1/(-x^2+1)^(1/2),y(0) = 0],y(x), singsol=all)
DSolve[{y'[x] == 1/(-x^2+1)^(1/2),y[0]==0},y[x],x,IncludeSingularSolutions -> True]