5.14 problem 14
Internal
problem
ID
[7955]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
5.0
Problem
number
:
14
Date
solved
:
Monday, October 21, 2024 at 04:39:26 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
Solve
\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\frac {1}{x} \end{align*}
Using series expansion around \(x=0\)
The type of the expansion point is first determined. This is done on the homogeneous part of
the ODE.
\[ y^{\prime \prime }+y^{\prime }+y = 0 \]
The following is summary of singularities for the above ode. Writing the ode as
\begin{align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end{align*}
Where
\begin{align*} p(x) &= 1\\ q(x) &= 1\\ \end{align*}
Table 120: Table \(p(x),q(x)\) singularites.
| |
| \(p(x)=1\) |
| |
|
singularity | type |
| |
| |
| \(q(x)=1\) |
| |
|
singularity | type |
| |
Combining everything together gives the following summary of singularities for the ode
as
Regular singular points : \([]\)
Irregular singular points : \([\infty ]\)
5.14.1 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+y^{\prime }+y=\frac {1}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +1=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-3}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=\mathit {C1} y_{1}\left (x \right )+\mathit {C2} y_{2}\left (x \right )+y_{p}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y={\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \mathit {C1} +{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \mathit {C2} +y_{p}\left (x \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (x \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (x \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (x \right )=-y_{1}\left (x \right ) \left (\int \frac {y_{2}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right )+y_{2}\left (x \right ) \left (\int \frac {y_{1}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right ), f \left (x \right )=\frac {1}{x}\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) & {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \\ -\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} & -\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\frac {\sqrt {3}\, {\mathrm e}^{-x}}{2} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-\frac {2 \sqrt {3}\, {\mathrm e}^{-\frac {x}{2}} \left (\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\int \frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{\frac {x}{2}}}{x}d x \right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\int \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{\frac {x}{2}}}{x}d x \right )\right )}{3} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=-\frac {\sqrt {3}\, \left (\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (-\frac {x \left (1+\mathrm {I} \sqrt {3}\right )}{2}\right )-\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (\frac {x \left (\mathrm {I} \sqrt {3}-1\right )}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{3} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y={\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \mathit {C2} +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \mathit {C1} -\frac {\sqrt {3}\, \left (\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (-\frac {x \left (1+\mathrm {I} \sqrt {3}\right )}{2}\right )-\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (\frac {x \left (\mathrm {I} \sqrt {3}-1\right )}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{3} \end {array} \]
5.14.2 Maple trace
Methods for second order ODEs:
5.14.3 Maple dsolve solution
Solving time : 0.019 (sec)
Leaf size : maple_leaf_size
dsolve(diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = 1/x,y(x),
series,x=0)
\[ \text {No solution found} \]
5.14.4 Mathematica DSolve solution
Solving time : 0.065 (sec)
Leaf size : 152
AsymptoticDSolveValue[{D[y[x],{x,2}]+D[y[x],x]+y[x]==1/x,{}},
y[x],{x,0,5}]
\[
y(x)\to c_2 x \left (-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x}{2}+1\right )+c_1 \left (\frac {x^3}{6}-\frac {x^2}{2}+1\right )+x \left (-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x}{2}+1\right ) \left (\frac {41 x^6}{4320}+\frac {x^5}{120}-\frac {x^4}{96}-\frac {x^3}{18}+x+\log (x)\right )+\left (\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \left (-\frac {x^6}{180}+\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^2}{4}-x\right )
\]