2.2.16 problem 16
Internal
problem
ID
[8246]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
16
Date
solved
:
Sunday, November 10, 2024 at 03:29:54 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
Solve
\begin{align*} y^{\prime \prime }-y^{\prime }-x y-x^{2}-1&=0 \end{align*}
Solved as second order Airy ode
Time used: 0.061 (sec)
This is Airy ODE. It has the general form
\[ a y^{\prime \prime } + b y^{\prime } + c x y = F(x) \]
Where in this case
\begin{align*} a &= 1\\ b &= -1\\ c &= -1\\ F &= x^{2}+1 \end{align*}
Therefore the solution to the homogeneous Airy ODE becomes
\[
y = c_1 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryAi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )+c_2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryBi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )
\]
Since this is inhomogeneous
Airy ODE, then we need to find the particular solution. The particular solution is now found
using the method of undetermined coefficients. Looking at the RHS of the ode, which is
\[ x^{2}+1 \]
Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial
solution is
\[ [\{1, x, x^{2}\}] \]
While the set of the basis functions for the homogeneous solution found earlier is
\[ \left \{{\mathrm e}^{\frac {x}{2}} \operatorname {AiryAi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right ), {\mathrm e}^{\frac {x}{2}} \operatorname {AiryBi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )\right \} \]
Since there is no duplication between the basis function in the UC_set and the basis
functions of the homogeneous solution, the trial solution is a linear combination of all the
basis in the UC_set.
\[
y_p = A_{3} x^{2}+A_{2} x +A_{1}
\]
The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) are found by substituting the above trial solution \(y_p\) into
the ODE and comparing coefficients. Substituting the trial solution into the ODE
and simplifying gives
\[
2 A_{3}-2 x A_{3}-A_{2}-x \left (A_{3} x^{2}+A_{2} x +A_{1}\right )-x^{2}-1 = 0
\]
Solving for the unknowns by comparing coefficients results
in
\[ [A_{1} = 0, A_{2} = -1, A_{3} = 0] \]
Substituting the above back in the above trial solution \(y_p\), gives the particular
solution
\[
y_p = -x
\]
Therefore the general solution is
\begin{align*}
y &= y_h + y_p \\
&= \left (c_1 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryAi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )+c_2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryBi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )\right ) + \left (-x\right ) \\
&= c_1 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryAi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )+c_2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryBi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )-x \\
\end{align*}
Will add steps showing solving for IC
soon.
Summary of solutions found
\begin{align*}
y &= c_1 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryAi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )+c_2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {AiryBi}\left (\left (-x -\frac {1}{4}\right ) \left (-1\right )^{{1}/{3}}\right )-x \\
\end{align*}
Maple step by step solution
Maple trace
`Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
trying high order exact linear fully integrable
trying differential order: 2; linear nonhomogeneous with symmetry [0,1]
trying a double symmetry of the form [xi=0, eta=F(x)]
-> Try solving first the homogeneous part of the ODE
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Trying a Liouvillian solution using Kovacics algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
-> Bessel
<- Bessel successful
<- special function solution successful
<- solving first the homogeneous part of the ODE successful`
Maple dsolve solution
Solving time : 0.021
(sec)
Leaf size : 28
dsolve(diff(diff(y(x),x),x)-diff(y(x),x)-x*y(x)-x^2-1 = 0,
y(x),singsol=all)
\[
y = {\mathrm e}^{\frac {x}{2}} \operatorname {AiryAi}\left (x +\frac {1}{4}\right ) c_{2} +{\mathrm e}^{\frac {x}{2}} \operatorname {AiryBi}\left (x +\frac {1}{4}\right ) c_{1} -x
\]
Mathematica DSolve solution
Solving time : 4.326
(sec)
Leaf size : 107
DSolve[{D[y[x],{x,2}]-D[y[x],x]-x*y[x]-x^2-1==0,{}},
y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{x/2} \left (\operatorname {AiryAi}\left (x+\frac {1}{4}\right ) \int _1^x-e^{-\frac {K[1]}{2}} \pi \operatorname {AiryBi}\left (K[1]+\frac {1}{4}\right ) \left (K[1]^2+1\right )dK[1]+\operatorname {AiryBi}\left (x+\frac {1}{4}\right ) \int _1^xe^{-\frac {K[2]}{2}} \pi \operatorname {AiryAi}\left (K[2]+\frac {1}{4}\right ) \left (K[2]^2+1\right )dK[2]+c_1 \operatorname {AiryAi}\left (x+\frac {1}{4}\right )+c_2 \operatorname {AiryBi}\left (x+\frac {1}{4}\right )\right )
\]