Internal problem ID [3684]
Internal file name [OUTPUT/3177_Sunday_June_05_2022_08_54_51_AM_93100744/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 15
Problem number: 430.
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]
\[ \boxed {y y^{\prime }-\sqrt {y^{2}-a^{2}}=0} \]
Integrating both sides gives \begin{align*} \int \frac {y}{\sqrt {-a^{2}+y^{2}}}d y &= \int d x \\ \sqrt {y^{2}-a^{2}}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {y^{2}-a^{2}} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \sqrt {y^{2}-a^{2}} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\sqrt {y^{2}-a^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {y^{2}-a^{2}}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y y^{\prime }}{\sqrt {y^{2}-a^{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y y^{\prime }}{\sqrt {y^{2}-a^{2}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \sqrt {y^{2}-a^{2}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {c_{1}^{2}+2 c_{1} x +a^{2}+x^{2}}, y=-\sqrt {c_{1}^{2}+2 c_{1} x +a^{2}+x^{2}}\right \} \end {array} \]
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: 29
dsolve(y(x)*diff(y(x),x) = sqrt(y(x)^2-a^2),y(x), singsol=all)
\[ x +\frac {\left (-y \left (x \right )+a \right ) \left (y \left (x \right )+a \right )}{\sqrt {y \left (x \right )^{2}-a^{2}}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.24 (sec). Leaf size: 51
DSolve[y[x] y'[x]==Sqrt[y[x]^2-a^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {a^2+(x+c_1){}^2} \\ y(x)\to \sqrt {a^2+(x+c_1){}^2} \\ y(x)\to -a \\ y(x)\to a \\ \end{align*}