5.25 problem 142

Internal problem ID [3398]
Internal file name [OUTPUT/2891_Sunday_June_05_2022_08_46_33_AM_4989097/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 5
Problem number: 142.
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]

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

5.25.1 Solving as quadrature ode

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

Summary

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

Verification of solutions

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

5.25.2 Maple step by step solution

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

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: 51

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

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

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 42

DSolve[x y'[x]==Sqrt[a^2-x^2],y[x],x,IncludeSingularSolutions -> True]
 

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