Consider the following equation: (3 − x^2 )y'' − 3xy' − y = 0
Derive the...
Consider the following equation: (3 − x^2 )y'' − 3xy' − y = 0
Derive the general solution of the given differential equation
about x = 0. Your answer should include a general formula for the
coefficents.
Diff. equations
Consider the equation (x^2 − 2)y''+ 3xy'+ y = 0.
a) Find the general solution as a power series centered at x =
0. Write the first six nonzero terms of the solution. And write the
solution using sigma notation with a formula for the coefficients.
Write the two linearly independent solutions that form the general
solution.
b) Find a power series solution satisfying the initial
conditions y(0) = 2 and y' (0) = 3. Write the first...
Consider the differential equation:
y'(x)+3xy+y^2=0.
y(1)=0. h=0.1
Solve the differential equation to determine y(1.3)
using:
a. Euler Method
b. Second order Taylor series method
c. Second order Runge Kutta method
d. Fourth order Runge-Kutta method
e. Heun’s predictor corrector method
f. Midpoint method
Consider the equation x^2+(y-2)^2 and the relation “(x, y) R (0,
2)”, where R is read as “has distance 1 of”. For example, “(0, 3) R
(0, 2)”, that is, “(0, 3) has distance 1 of (0, 2)”. This relation
can also be read as “(x, y) belongs to the circle of radius 1 with
center (0, 2)”. In other words: “(x, y) satisfies this equation if,
and only if, (x, y) R (0, 2)”. Does this equation determine a...
Consider a Cauchy-Euler equation x^2y''- xy' +y =x^3 for
x>0.
a) Rewrite the equation as constant- coefficeint equation by
substituting x = e^t.
b) Solve it when x(1)=0, x'(1)=1.
Consider the differential equation
(x
2 + 1)y
′′ − 4xy′ + 6y = 0.
(a) Determine all singular points and find a minimum value for the
radius of convergence of
a power series solution about x0 = 0.
(b) Use a power series expansion y(x) = ∑∞
n=0
anx
n
about the ordinary point x0 = 0, to find
a general solution to the above differential equation, showing all
necessary steps including the
following:
(i) recurrence relation;
(ii) determination...
Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are
all real numbers
(a) Derive an equation for a line tangent to both of these
parabolas (show all steps with a proof, assuming that such
a line exists)
(b) Assume that the doubly-tangent line has an equation y+Ax+B.
Find an example of values of a,b,c (other than the ones given here)
such that A,B ∈ Z