2.5.12 problem 12
Internal
problem
ID
[8399]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
5.0
Problem
number
:
12
Date
solved
:
Sunday, November 10, 2024 at 03:44:24 AM
CAS
classification
:
[[_2nd_order, _missing_y]]
Solve
\begin{align*} y^{\prime \prime }+y^{\prime }&=\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.
\[ y^{\prime \prime }+y^{\prime } = 0 \]
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) &= 0\\ \end{align*}
Table 2.118: Table \(p(x),q(x)\) singularites.
| |
\(p(x)=1\) |
| |
singularity | type |
| |
| |
\(q(x)=0\) |
| |
singularity | type |
| |
Combining everything together gives the following summary of singularities for the ode
as
Regular singular points : \([]\)
Irregular singular points : \([\infty ]\)
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {d}{d x}y \left (x \right )=\frac {1}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r =0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & r \left (r +1\right )=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-1, 0\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-x} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )=1 \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\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 \left (x \right )=\mathit {C1} \,{\mathrm e}^{-x}+\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}^{-x} & 1 \\ -{\mathrm e}^{-x} & 0 \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )={\mathrm e}^{-x} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-{\mathrm e}^{-x} \left (\int \frac {{\mathrm e}^{x}}{x}d x \right )+\int \frac {1}{x}d x \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )={\mathrm e}^{-x} \mathrm {Ei}_{1}\left (-x \right )+\ln \left (x \right ) \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-x}+\mathit {C2} +{\mathrm e}^{-x} \mathrm {Ei}_{1}\left (-x \right )+\ln \left (x \right ) \end {array} \]
Maple trace
`Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
trying high order exact linear fully integrable
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(_b(_a)*_a-1)/_a, _b(_a)` *** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
<- high order exact linear fully integrable successful`
Maple dsolve solution
Solving time : 0.042
(sec)
Leaf size : maple_leaf_size
dsolve(diff(y(x),x)+diff(diff(y(x),x),x) = 1/x,y(x),
series,x=0)
\[ \text {No solution found} \]
Mathematica DSolve solution
Solving time : 0.024
(sec)
Leaf size : 159
AsymptoticDSolveValue[{D[y[x],{x,2}]+D[y[x],x]==1/x,{}},
y[x],{x,0,5}]
\[
y(x)\to -\frac {x^6}{4320}-\frac {x^5}{600}-\frac {x^4}{96}-\frac {x^3}{18}-\frac {x^2}{4}+c_2 \left (-\frac {x^5}{720}+\frac {x^4}{120}-\frac {x^3}{24}+\frac {x^2}{6}-\frac {x}{2}+1\right ) x+\left (-\frac {x^5}{720}+\frac {x^4}{120}-\frac {x^3}{24}+\frac {x^2}{6}-\frac {x}{2}+1\right ) x \left (\frac {x^6}{2160}+\frac {x^5}{600}+\frac {x^4}{96}+\frac {x^3}{18}+\frac {x^2}{4}+x+\log (x)\right )-x+c_1
\]