2.5.10 Problem 10
Internal
problem
ID
[10245]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
5.0
Problem
number
:
10
Date
solved
:
Thursday, November 27, 2025 at 10:27:38 AM
CAS
classification
:
[_quadrature]
\begin{align*}
y^{\prime }&=\frac {1}{x} \\
\end{align*}
Series expansion around
\(x=0\).
\begin{align*}
y^{\prime }&=\frac {1}{x} \\
\end{align*}
Series expansion around
\(x=0\).
Since this is an inhomogeneous, then let the solution be
\[ y = y_h + y_p \]
Where
\(y_h\) is the solution to the
homogeneous ode
\(y^{\prime } = 0\),and
\(y_p\) is a particular solution to the inhomogeneous ode. First, we solve for
\(y_h\) Let
the homogeneous solution be represented as Frobenius power series of the form
\[
y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}
\]
Then
\[
y^{\prime } = \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}
\]
Substituting
the above back into the ode gives
\begin{equation}
\tag{1} \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} = 0
\end{equation}
Which simplifies to
\begin{equation}
\tag{2A} \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} = 0
\end{equation}
The next step is to make all powers of
\(x\) be
\(n +r -1\)
in each summation term. Going over each summation term above with power of
\(x\) in it which is not
already
\(x^{n +r -1}\) and adjusting the power and the corresponding index gives Substituting all the
above in Eq (2A) gives the following equation where now all powers of
\(x\) are the same
and equal to
\(n +r -1\).
\begin{equation}
\tag{2B} \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} = 0
\end{equation}
The indicial equation is obtained from
\(n=0\). From Eq (2) this gives
\[
\left (n +r \right ) a_{n} x^{n +r -1} = 0
\]
When
\(n=0\)
the above becomes
\[
r a_{0} x^{-1+r} = 0
\]
The corresponding balance equation is found by replacing
\(r\) by
\(m\)
and
\(a\) by
\(c\) to avoid confusing terms between particular solution and the homogeneous
solution. Hence the balance equation is
\[
m c_{0} x^{-1+m} = \frac {1}{x}
\]
This equation will used later to find the particular
solution.
Since \(a_{0}\neq 0\) then the indicial equation becomes
\[
r \,x^{-1+r} = 0
\]
Since the above is true for all
\(x\) then the indicial equation
simplifies to
\[
r = 0
\]
Solving for
\(r\) gives the root of the indicial equation as
\[ r=0 \]
From the above we see that there is no recurrence relation since there is only one summation term.
Therefore all \(a_{n}\) terms are zero except for \(a_{0}\). Hence
\begin{align*} y_h &= a_{0} \left (1+O\left (x^{6}\right )\right ) \end{align*}
Now the particular solution is found Unable to solve the balance equation \(m c_{0} x^{-1+m} = \frac {1}{x}\) for \(c_{0}\) and \(x\). No particular
solution exists.
Unable to find the particular solution. No solution exist.
✗ Maple
Order:=6;
ode:=diff(y(x),x) = 1/x;
dsolve(ode,y(x),type='series',x=0);
\[ \text {No solution found} \]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
<- quadrature successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int \frac {1}{x}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\ln \left (x \right )+\mathit {C1} \end {array} \]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 8
ode=D[y[x],x]==1/x;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\[
y(x)\to \log (x)+c_1
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - 1/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
ValueError : ODE Derivative(y(x), x) - 1/x does not match hint 1st_power_series