Internal problem ID [6125]
Internal file name [OUTPUT/5373_Sunday_June_05_2022_03_35_39_PM_21493951/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 1.2 THE NATURE OF SOLUTIONS.
Page 9
Problem number: 2(g).
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]
\[ \boxed {y^{\prime }=\arcsin \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \arcsin \left (x \right )\,\mathop {\mathrm {d}x}}\\ &= x \arcsin \left (x \right )+\sqrt {-x^{2}+1}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x \arcsin \left (x \right )+\sqrt {-x^{2}+1}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = x \arcsin \left (x \right )+\sqrt {-x^{2}+1}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\arcsin \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \arcsin \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x \arcsin \left (x \right )+\sqrt {-x^{2}+1}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x \arcsin \left (x \right )+\sqrt {-x^{2}+1}+c_{1} \end {array} \]
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: 19
dsolve(diff(y(x),x)=arcsin(x),y(x), singsol=all)
\[ y \left (x \right ) = x \arcsin \left (x \right )+\sqrt {-x^{2}+1}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 23
DSolve[y'[x]==ArcSin[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \arcsin (x)+\sqrt {1-x^2}+c_1 \]