20.10.8 problem Problem 21
Internal
problem
ID
[3780]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
8,
Linear
differential
equations
of
order
n.
Section
8.8,
A
Differential
Equation
with
Nonconstant
Coefficients.
page
567
Problem
number
:
Problem
21
Date
solved
:
Sunday, March 30, 2025 at 02:08:21 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} y^{\prime \prime }-\left (2 m -1\right ) x y^{\prime }+m^{2} y&=x^{m} \ln \left (x \right )^{k} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 33
ode:=x^2*diff(diff(y(x),x),x)-(2*m-1)*x*diff(y(x),x)+m^2*y(x) = x^m*ln(x)^k;
dsolve(ode,y(x), singsol=all);
\[
y = x^{m} \left (c_2 +\ln \left (x \right ) c_1 +\frac {\ln \left (x \right )^{k} \ln \left (x \right )^{2}}{k^{2}+3 k +2}\right )
\]
✓ Mathematica. Time used: 0.057 (sec). Leaf size: 35
ode=x^2*D[y[x],{x,2}]-(2*m-1)*x*D[y[x],x]+m^2*y[x]==x^m*(Log[x])^k;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to x^m \left (\frac {\log ^{k+2}(x)}{k^2+3 k+2}+c_2 m \log (x)+c_1\right )
\]
✓ Sympy. Time used: 3.378 (sec). Leaf size: 121
from sympy import *
x = symbols("x")
k = symbols("k")
m = symbols("m")
y = Function("y")
ode = Eq(m**2*y(x) + x**2*Derivative(y(x), (x, 2)) - x*(2*m - 1)*Derivative(y(x), x) - x**m*log(x)**k,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = x^{\operatorname {re}{\left (m\right )}} \left (C_{3} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )} + C_{4} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (m\right )} \right )} + \left (C_{1} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )} + C_{2} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (m\right )} \right )}\right ) \log {\left (x \right )} + \left (\log {\left (x \right )} \int \frac {x^{m - \operatorname {re}{\left (m\right )} - 1} \log {\left (x \right )}^{k}}{\sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )}}\, dx - \int \frac {x^{m - \operatorname {re}{\left (m\right )} - 1} \log {\left (x \right )}^{k + 1}}{\sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )}}\, dx\right ) \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )}\right )
\]