15.21 problem 429

Internal problem ID [3683]
Internal file name [OUTPUT/3176_Sunday_June_05_2022_08_54_50_AM_25505876/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 15
Problem number: 429.
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]

\[ \boxed {y y^{\prime }-\sqrt {y^{2}+a^{2}}=0} \]

15.21.1 Solving as quadrature ode

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*}

Summary

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.

15.21.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\sqrt {y^{2}+a^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {y^{2}+a^{2}}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y y^{\prime }}{\sqrt {y^{2}+a^{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y y^{\prime }}{\sqrt {y^{2}+a^{2}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \sqrt {y^{2}+a^{2}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {c_{1}^{2}+2 c_{1} x -a^{2}+x^{2}}, y=-\sqrt {c_{1}^{2}+2 c_{1} x -a^{2}+x^{2}}\right \} \end {array} \]

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: 18

dsolve(y(x)*diff(y(x),x) = sqrt(y(x)^2+a^2),y(x), singsol=all)
 

\[ x -\sqrt {y \left (x \right )^{2}+a^{2}}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.223 (sec). Leaf size: 61

DSolve[y[x] y'[x]==Sqrt[a^2+y[x]^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 -i a \\ y(x)\to i a \\ \end{align*}