Internal problem ID [11156]
Internal file name [OUTPUT/10142_Sunday_November_27_2022_04_33_53_PM_41838316/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 16.
Integrating factors by inspection. Page 23
Problem number: Ex 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "exactWithIntegrationFactor"
Maple gives the following as the ode type
[`y=_G(x,y')`]
\[ \boxed {\frac {y^{\prime } x -y}{\sqrt {-y^{2}+x^{2}}}-y^{\prime } x=0} \]
Entering Exact first order ODE solver. (Form one type)
To solve an ode of the form\begin {equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A} \end {equation} We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \] Hence\begin {equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B} \end {equation} Comparing (A,B) shows that\begin {align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end {align*}
But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\] If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is \[ M(x,y) \mathop {\mathrm {d}x}+ N(x,y) \mathop {\mathrm {d}y}=0 \tag {1A} \] Therefore \begin {align*} \left (\frac {x}{\sqrt {x^{2}-y^{2}}}-x\right )\mathop {\mathrm {d}y} &= \left (\frac {y}{\sqrt {x^{2}-y^{2}}}\right )\mathop {\mathrm {d}x}\\ \left (-\frac {y}{\sqrt {x^{2}-y^{2}}}\right )\mathop {\mathrm {d}x} + \left (\frac {x}{\sqrt {x^{2}-y^{2}}}-x\right )\mathop {\mathrm {d}y} &= 0 \tag {2A} \end {align*}
Comparing (1A) and (2A) shows that \begin {align*} M(x,y) &= -\frac {y}{\sqrt {x^{2}-y^{2}}}\\ N(x,y) &= \frac {x}{\sqrt {x^{2}-y^{2}}}-x \end {align*}
The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied \[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial x} \] Using result found above gives \begin {align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (-\frac {y}{\sqrt {x^{2}-y^{2}}}\right )\\ &= -\frac {x^{2}}{\left (x^{2}-y^{2}\right )^{\frac {3}{2}}} \end {align*}
And \begin {align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (\frac {x}{\sqrt {x^{2}-y^{2}}}-x\right )\\ &= -\frac {\left (x^{2}-y^{2}\right )^{\frac {3}{2}}+y^{2}}{\left (x^{2}-y^{2}\right )^{\frac {3}{2}}} \end {align*}
Since \(\frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let \begin {align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial y} - \frac {\partial N}{\partial x} \right ) \\ &=-\frac {\sqrt {x^{2}-y^{2}}}{x \left (\sqrt {x^{2}-y^{2}}-1\right )}\left ( \left ( -\frac {1}{\sqrt {x^{2}-y^{2}}}-\frac {y^{2}}{\left (x^{2}-y^{2}\right )^{\frac {3}{2}}}\right ) - \left (\frac {1}{\sqrt {x^{2}-y^{2}}}-\frac {x^{2}}{\left (x^{2}-y^{2}\right )^{\frac {3}{2}}}-1 \right ) \right ) \\ &=-\frac {1}{x} \end {align*}
Since \(A\) does not depend on \(y\), then it can be used to find an integrating factor. The integrating factor \(\mu \) is \begin {align*} \mu &= e^{ \int A \mathop {\mathrm {d}x} } \\ &= e^{\int -\frac {1}{x}\mathop {\mathrm {d}x} } \end {align*}
The result of integrating gives \begin {align*} \mu &= e^{-\ln \left (x \right ) } \\ &= \frac {1}{x} \end {align*}
\(M\) and \(N\) are multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\) for now so not to confuse them with the original \(M\) and \(N\). \begin {align*} \overline {M} &=\mu M \\ &= \frac {1}{x}\left (-\frac {y}{\sqrt {x^{2}-y^{2}}}\right ) \\ &= -\frac {y}{\sqrt {x^{2}-y^{2}}\, x} \end {align*}
And \begin {align*} \overline {N} &=\mu N \\ &= \frac {1}{x}\left (\frac {x}{\sqrt {x^{2}-y^{2}}}-x\right ) \\ &= -1+\frac {1}{\sqrt {x^{2}-y^{2}}} \end {align*}
Now a modified ODE is ontained from the original ODE, which is exact and can be solved. The modified ODE is \begin {align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}} &= 0 \\ \left (-\frac {y}{\sqrt {x^{2}-y^{2}}\, x}\right ) + \left (-1+\frac {1}{\sqrt {x^{2}-y^{2}}}\right ) \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}} &= 0 \end {align*}
The following equations are now set up to solve for the function \(\phi \left (x,y\right )\) \begin {align*} \frac {\partial \phi }{\partial x } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial y } &= \overline {N}\tag {2} \end {align*}
Integrating (1) w.r.t. \(x\) gives \begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {M}\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int -\frac {y}{\sqrt {x^{2}-y^{2}}\, x}\mathop {\mathrm {d}x} \\ \tag{3} \phi &= \frac {y \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}}{x}\right )\right )}{\sqrt {-y^{2}}}+ f(y) \\ \end{align*} Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives \begin{align*} \tag{4} \frac {\partial \phi }{\partial y} &= \frac {\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}}{x}\right )}{\sqrt {-y^{2}}}+\frac {y^{2} \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}}{x}\right )\right )}{\left (-y^{2}\right )^{\frac {3}{2}}}+\frac {y \left (-\frac {\sqrt {x^{2}-y^{2}}\, y}{\sqrt {-y^{2}}}-\frac {\sqrt {-y^{2}}\, y}{\sqrt {x^{2}-y^{2}}}-2 y \right )}{\sqrt {-y^{2}}\, \left (\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}\right )}+f'(y) \\ &=-\frac {-2 y^{2}+2 \sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+x^{2}}{\sqrt {x^{2}-y^{2}}\, \left (-\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+y^{2}\right )}+f'(y) \\ \end{align*} But equation (2) says that \(\frac {\partial \phi }{\partial y} = -1+\frac {1}{\sqrt {x^{2}-y^{2}}}\). Therefore equation (4) becomes \begin{equation} \tag{5} -1+\frac {1}{\sqrt {x^{2}-y^{2}}} = -\frac {-2 y^{2}+2 \sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+x^{2}}{\sqrt {x^{2}-y^{2}}\, \left (-\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+y^{2}\right )}+f'(y) \end{equation} Solving equation (5) for \( f'(y)\) gives \begin{align*} f'(y) &= -\frac {-\sqrt {x^{2}-y^{2}}\, y^{2}+\sqrt {-y^{2}}\, x^{2}-y^{2} \sqrt {-y^{2}}+\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+x^{2}-y^{2}}{\sqrt {x^{2}-y^{2}}\, \left (\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}\right )} \\ &= \frac {\left (-y^{2}+\sqrt {-y^{2}}\right ) \sqrt {x^{2}-y^{2}}+\left (x -y \right ) \left (x +y \right ) \left (\sqrt {-y^{2}}+1\right )}{\sqrt {x^{2}-y^{2}}\, \left (-\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+y^{2}\right )}\\ \end{align*} Integrating the above w.r.t \(y\) results in \begin{align*} \int f'(y) \mathop {\mathrm {d}y} &= \int \left ( \frac {\left (-y^{2}+\sqrt {-y^{2}}\right ) \sqrt {x^{2}-y^{2}}+\left (x -y \right ) \left (x +y \right ) \left (\sqrt {-y^{2}}+1\right )}{\sqrt {x^{2}-y^{2}}\, \left (-\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}+y^{2}\right )}\right ) \mathop {\mathrm {d}y} \\ f(y) &= -y +\frac {\sqrt {-y^{2}}\, \ln \left (y \right )}{y}+ c_{1} \\ \end{align*} Where \(c_{1}\) is constant of integration. Substituting result found above for \(f(y)\) into equation (3) gives \(\phi \) \[ \phi = \frac {y \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}}{x}\right )\right )}{\sqrt {-y^{2}}}-y +\frac {\sqrt {-y^{2}}\, \ln \left (y \right )}{y}+ c_{1} \] But since \(\phi \) itself is a constant function, then let \(\phi =c_{2}\) where \(c_{2}\) is new constant and combining \(c_{1}\) and \(c_{2}\) constants into new constant \(c_{1}\) gives the solution as \[ c_{1} = \frac {y \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {x^{2}-y^{2}}-y^{2}}{x}\right )\right )}{\sqrt {-y^{2}}}-y +\frac {\sqrt {-y^{2}}\, \ln \left (y \right )}{y} \]
The solution(s) found are the following \begin{align*} \tag{1} \frac {y \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {-y^{2}+x^{2}}-y^{2}}{x}\right )\right )}{\sqrt {-y^{2}}}-y+\frac {\sqrt {-y^{2}}\, \ln \left (y\right )}{y} &= c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {y \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {-y^{2}+x^{2}}-y^{2}}{x}\right )\right )}{\sqrt {-y^{2}}}-y+\frac {\sqrt {-y^{2}}\, \ln \left (y\right )}{y} = c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } x -y}{\sqrt {-y^{2}+x^{2}}}-y^{\prime } x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y}{\sqrt {-y^{2}+x^{2}}\, \left (\frac {x}{\sqrt {-y^{2}+x^{2}}}-x \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5`[0, (x^2-y^2)^(1/2)/((x^2-y^2)^(1/2)-1)]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 27
dsolve((x*diff(y(x),x)-y(x))/sqrt(x^2-y(x)^2)=x*diff(y(x),x),y(x), singsol=all)
\[ y \left (x \right )-\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.896 (sec). Leaf size: 29
DSolve[(x*y'[x]-y[x])/Sqrt[x^2-y[x]^2]==x*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\arctan \left (\frac {\sqrt {x^2-y(x)^2}}{y(x)}\right )+y(x)=c_1,y(x)\right ] \]