Internal problem ID [14078]
Internal file name [OUTPUT/13759_Saturday_March_02_2024_02_49_18_PM_84717721/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 42.
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]
\[ \boxed {y^{\prime }=\frac {\sqrt {x^{2}-16}}{x}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {\sqrt {x^{2}-16}}{x}\,\mathop {\mathrm {d}x}}\\ &= \sqrt {x^{2}-16}-4 \arctan \left (\frac {\sqrt {x^{2}-16}}{4}\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {x^{2}-16}-4 \arctan \left (\frac {\sqrt {x^{2}-16}}{4}\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {x^{2}-16}-4 \arctan \left (\frac {\sqrt {x^{2}-16}}{4}\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {x^{2}-16}}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sqrt {x^{2}-16}}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\sqrt {x^{2}-16}+4 \arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sqrt {x^{2}-16}+4 \arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )+c_{1} \end {array} \]
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: 25
dsolve(diff(y(x),x)=sqrt(x^2-16)/x,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x^{2}-16}+4 \arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 33
DSolve[y'[x]==Sqrt[x^2-16]/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -4 \arctan \left (\frac {\sqrt {x^2-16}}{4}\right )+\sqrt {x^2-16}+c_1 \]