Internal problem ID [8426]
Internal file name [OUTPUT/7359_Sunday_June_05_2022_10_53_42_PM_89137119/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 89.
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]
\[ \boxed {x y^{\prime }=\sqrt {a^{2}-x^{2}}} \]
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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }=\sqrt {a^{2}-x^{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {a^{2}-x^{2}}}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sqrt {a^{2}-x^{2}}}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\sqrt {a^{2}-x^{2}}-\frac {a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-x^{2}}}{x}\right )}{\sqrt {a^{2}}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {-a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-x^{2}}}{x}\right )+\sqrt {a^{2}}\, \sqrt {a^{2}-x^{2}}+c_{1} \sqrt {a^{2}}}{\sqrt {a^{2}}} \end {array} \]
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)=0,y(x), singsol=all)
\[ y \left (x \right ) = -a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-x^{2}}+a \right )}{x}\right )+\sqrt {a^{2}-x^{2}}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 42
DSolve[x*y'[x] - Sqrt[a^2 - x^2]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -a \text {arctanh}\left (\frac {\sqrt {a^2-x^2}}{a}\right )+\sqrt {a^2-x^2}+c_1 \]