77.17.9 problem 9
Internal
problem
ID
[20490]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
III.
Ordinary
linear
differential
equations
with
constant
coefficients.
Misc.
Examples
on
chapter
III
at
page
50
Problem
number
:
9
Date
solved
:
Thursday, October 02, 2025 at 06:03:29 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }+16 y&=16 x^{2}+256 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 57
ode:=diff(diff(diff(diff(y(x),x),x),x),x)+diff(diff(y(x),x),x)+16*y(x) = 16*x^2+256;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (8 c_1 \cos \left (\frac {3 x}{2}\right )+8 c_2 \sin \left (\frac {3 x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {7}\, x}{2}}}{8}+\frac {\left (8 c_3 \cos \left (\frac {3 x}{2}\right )+8 c_4 \sin \left (\frac {3 x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {7}\, x}{2}}}{8}+x^{2}+\frac {127}{8}
\]
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 132
ode=D[y[x],{x,4}]+D[y[x],{x,2}]+16*y[x]==16*x^2+256;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-2 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )} \cos \left (2 x \cos \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )\right ) \left (c_3 e^{4 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )}+c_2\right )+e^{-2 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )} \left (c_1 e^{4 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )}+c_4\right ) \sin \left (2 x \cos \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )\right )+x^2+\frac {127}{8} \end{align*}
✓ Sympy. Time used: 0.159 (sec). Leaf size: 117
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-16*x**2 + 16*y(x) + Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)) - 256,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = x^{2} + \left (C_{1} \sin {\left (2 x \cos {\left (\frac {\operatorname {atan}{\left (3 \sqrt {7} \right )}}{2} \right )} \right )} + C_{2} \cos {\left (2 x \cos {\left (\frac {\operatorname {atan}{\left (3 \sqrt {7} \right )}}{2} \right )} \right )}\right ) e^{- 2 x \sin {\left (\frac {\operatorname {atan}{\left (3 \sqrt {7} \right )}}{2} \right )}} + \left (C_{3} \sin {\left (2 x \cos {\left (\frac {\operatorname {atan}{\left (3 \sqrt {7} \right )}}{2} \right )} \right )} + C_{4} \cos {\left (2 x \cos {\left (\frac {\operatorname {atan}{\left (3 \sqrt {7} \right )}}{2} \right )} \right )}\right ) e^{2 x \sin {\left (\frac {\operatorname {atan}{\left (3 \sqrt {7} \right )}}{2} \right )}} + \frac {127}{8}
\]