3.14 problem 24 (g)

Internal problem ID [5275]
Internal file name [OUTPUT/4766_Friday_February_02_2024_05_11_44_AM_93727827/index.tex]

Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section: Chapter 5. Equations of first order and first degree (Exact equations). Supplemetary problems. Page 33
Problem number: 24 (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]

\[ \boxed {-\sqrt {a^{2}-x^{2}}\, y^{\prime }=-1} \]

3.14.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \frac {1}{\sqrt {a^{2}-x^{2}}}\,\mathop {\mathrm {d}x}}\\ &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ \end{align*}

Verification of solutions

\[ y = \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \] Verified OK.

3.14.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\sqrt {a^{2}-x^{2}}\, y^{\prime }=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{\sqrt {a^{2}-x^{2}}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{\sqrt {a^{2}-x^{2}}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \end {array} \]

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: 20

dsolve(1-(sqrt(a^2-x^2))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 24

DSolve[1-(Sqrt[a^2-x^2])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
                                                                                    
                                                                                    
 

\[ y(x)\to \arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1 \]