Internal problem ID [4769]
Internal file name [OUTPUT/4262_Sunday_June_05_2022_12_49_35_PM_96920737/index.tex]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 3. Linear First-Order Equations. page
403
Problem number: 9.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-x y=2 x \sqrt {-x^{2}+1}} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=\frac {x}{x^{2}-1}\\ q(x) &=-\frac {2 x \sqrt {-x^{2}+1}}{x^{2}-1} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {x y}{x^{2}-1} = -\frac {2 x \sqrt {-x^{2}+1}}{x^{2}-1} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {x}{x^{2}-1}d x} \\ &= {\mathrm e}^{\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}} \\ \end{align*} Which simplifies to \[ \mu = \sqrt {x -1}\, \sqrt {x +1} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (-\frac {2 x \sqrt {-x^{2}+1}}{x^{2}-1}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\sqrt {x -1}\, \sqrt {x +1}\, y\right ) &= \left (\sqrt {x -1}\, \sqrt {x +1}\right ) \left (-\frac {2 x \sqrt {-x^{2}+1}}{x^{2}-1}\right )\\ \mathrm {d} \left (\sqrt {x -1}\, \sqrt {x +1}\, y\right ) &= \left (\frac {2 \sqrt {x -1}\, \sqrt {x +1}\, x}{\sqrt {-x^{2}+1}}\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \sqrt {x -1}\, \sqrt {x +1}\, y &= \int {\frac {2 \sqrt {x -1}\, \sqrt {x +1}\, x}{\sqrt {-x^{2}+1}}\,\mathrm {d} x}\\ \sqrt {x -1}\, \sqrt {x +1}\, y &= \frac {x^{2} \sqrt {x -1}\, \sqrt {x +1}}{\sqrt {-x^{2}+1}} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\sqrt {x -1}\, \sqrt {x +1}\) results in \begin {align*} y &= \frac {x^{2}}{\sqrt {-x^{2}+1}}+\frac {c_{1}}{\sqrt {x -1}\, \sqrt {x +1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2}}{\sqrt {-x^{2}+1}}+\frac {c_{1}}{\sqrt {x -1}\, \sqrt {x +1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2}}{\sqrt {-x^{2}+1}}+\frac {c_{1}}{\sqrt {x -1}\, \sqrt {x +1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (-x^{2}+1\right ) y^{\prime }-x y=2 x \sqrt {-x^{2}+1} \\ \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 {x y+2 x \sqrt {-x^{2}+1}}{-x^{2}+1} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x y}{x^{2}-1}+\frac {2 x}{\sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {x y}{x^{2}-1}=\frac {2 x}{\sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {x y}{x^{2}-1}\right )=\frac {2 \mu \left (x \right ) x}{\sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {x y}{x^{2}-1}\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=\frac {\mu \left (x \right ) x}{x^{2}-1} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\sqrt {x -1}\, \sqrt {x +1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int \frac {2 \mu \left (x \right ) x}{\sqrt {-x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \frac {2 \mu \left (x \right ) x}{\sqrt {-x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \frac {2 \mu \left (x \right ) x}{\sqrt {-x^{2}+1}}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\sqrt {x -1}\, \sqrt {x +1} \\ {} & {} & y=\frac {\int \frac {2 \sqrt {x -1}\, \sqrt {x +1}\, x}{\sqrt {-x^{2}+1}}d x +c_{1}}{\sqrt {x -1}\, \sqrt {x +1}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\frac {x^{2} \sqrt {x -1}\, \sqrt {x +1}}{\sqrt {-x^{2}+1}}+c_{1}}{\sqrt {x -1}\, \sqrt {x +1}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 30
dsolve((1-x^2)*diff(y(x),x)=x*y(x)+2*x*sqrt(1-x^2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{2}}{\sqrt {-x^{2}+1}}+\frac {c_{1}}{\sqrt {x -1}\, \sqrt {1+x}} \]
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 33
DSolve[(1-x^2)*y'[x]==x*y[x]+2*x*Sqrt[1-x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^2}{\sqrt {1-x^2}}+\frac {c_1}{\sqrt {x^2-1}} \]