Internal problem ID [1797]
Internal file name [OUTPUT/1798_Sunday_June_05_2022_02_31_51_AM_84290585/index.tex]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number: 5.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (-t^{2}+1\right ) y^{\prime \prime }+\frac {y^{\prime }}{\sin \left (t +1\right )}+y=0} \] With the expansion point for the power series method at \(t = -1\).
The ode does not have its expansion point at \(t = 0\), therefore to simplify the computation of power series expansion, change of variable is made on the independent variable to shift the initial conditions and the expasion point back to zero. The new ode is then solved more easily since the expansion point is now at zero. The solution converted back to the original independent variable. Let \[ x = t +1 \] The ode is converted to be in terms of the new independent variable \(x\). This results in \[ \left (1-\left (x -1\right )^{2}\right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\csc \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right ) = 0 \] With its expansion point and initial conditions now at \(x = 0\). The transformed ODE is now solved. The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ \left (-x^{2}+2 x \right ) \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+\csc \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right ) = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} \frac {d^{2}}{d x^{2}}y \left (x \right )+p(x) \frac {d}{d x}y \left (x \right ) + q(x) y \left (x \right ) &=0 \end {align*}
Where \begin {align*} p(x) &= -\frac {\csc \left (x \right )}{x \left (-2+x \right )}\\ q(x) &= -\frac {1}{\left (-2+x \right ) x}\\ \end {align*}
| \(p(x)=-\frac {\csc \left (x \right )}{x \left (-2+x \right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``irregular''}\) |
| \(x = 2\) | \(\text {``regular''}\) |
| \(x = Z \pi \) | \(\text {``regular''}\) |
| \(q(x)=-\frac {1}{\left (-2+x \right ) x}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 2\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([2, Z \pi ]\)
Irregular singular points : \([0, \infty ]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating. Unable to solve the transformed ode. Terminating.
Verification of solutions N/A
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients -> trying with_periodic_functions in the coefficients --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 5 <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying differential order: 2; exact nonlinear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients -> trying with_periodic_functions in the coefficients trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients -> trying with_periodic_functions in the coefficients trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients -> trying with_periodic_functions in the coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying with_periodic_functions in the coefficients --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 5 --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
Order:=6; dsolve((1-t^2)*diff(y(t),t$2)+1/sin(t+1)*diff(y(t),t)+y(t)=0,y(t),type='series',t=-1);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 111
AsymptoticDSolveValue[(1-t^2)*y''[t]+1/Sin[t+1]*y'[t]+y[t]==0,y[t],{t,-1,5}]
\[ y(t)\to c_2 e^{\frac {1}{2 (t+1)}} \left (\frac {516353141702117 (t+1)^5}{33443020800}+\frac {53349163853 (t+1)^4}{39813120}+\frac {58276991 (t+1)^3}{414720}+\frac {21397 (t+1)^2}{1152}+\frac {79 (t+1)}{24}+1\right ) (t+1)^{7/4}+c_1 \left (\frac {53}{5} (t+1)^5-\frac {25}{12} (t+1)^4+\frac {2}{3} (t+1)^3-\frac {1}{2} (t+1)^2+1\right ) \]