Internal problem ID [8396]
Internal file name [OUTPUT/7329_Sunday_June_05_2022_05_50_44_PM_10415092/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 59.
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]
\[ \boxed {y^{\prime }-a \sqrt {y^{2}+1}=b} \]
Integrating both sides gives \begin {align*} \int \frac {1}{a \sqrt {y^{2}+1}+b}d y &= \int {dx}\\ \int _{}^{y}\frac {1}{a \sqrt {\textit {\_a}^{2}+1}+b}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{a \sqrt {\textit {\_a}^{2}+1}+b}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{a \sqrt {\textit {\_a}^{2}+1}+b}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 26
dsolve(diff(y(x),x) - a*sqrt(y(x)^2+1) - b=0,y(x), singsol=all)
\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{a \sqrt {\textit {\_a}^{2}+1}+b}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.624 (sec). Leaf size: 127
DSolve[y'[x] - a*Sqrt[y[x]^2+1] - b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\frac {2 b \arctan \left (\frac {\left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right ) a+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1] \\ y(x)\to -\frac {\sqrt {b^2-a^2}}{a} \\ y(x)\to \frac {\sqrt {b^2-a^2}}{a} \\ \end{align*}