Question

Show that f(x) = C_{1}e^{4x} +
C_{2}e^{-2x} is a solution to the differential
equation: y’’ – 2y’ – 8y = 0, for all constants C_{1} and
C_{2}. Then find values for C_{1} and C_{2}
such that y(0) = 1 and y’(0) = 0.

Answer #1

The general solution of the equation
y′′+6y′+13y=0 is
y=c1e-3xcos(2x)+c2e−3xsin(2x)
Find values of c1 and c2 so that y(0)=1
and y′(0)=−9.
c1=?
c2=?
Plug these values into the general solution to obtain
the unique solution.
y=?

Suppose x=c1e−t+c2e^5t. Verify that x=c1e^−t+c2e^5t is a
solution to x′′−4x′−5x=0 by substituting it into the differential
equation. (Enter the terms in the order given. Enter c1 as c1 and
c2 as c2.)

Consider the differential equation: .
Let y = f(x) be the particular solution to the differential
equation with initial condition, f(0) = -1.
Part (a) Find . Show or explain your work, do not
just give an answer.

Solve the following differential equation by variation of
parameters. Fully evaluate all integrals.
y′′+9y=sec(3x).
a. Find the most general solution to the associated homogeneous
differential equation. Use c1 and c2 in your
answer to denote arbitrary constants, and enter them as c1 and
c2.
b. Find a particular solution to the nonhomogeneous differential
equation y′′+9y=sec(3x).
c. Find the most general solution to the original nonhomogeneous
differential equation. Use c1 and c2 in your
answer to denote arbitrary constants.

Consider the differential equation dy/dx= 2y(x+1)
a) sketch a slope field
b) Show that any point with initial condition x = –1 in the 2nd
quadrant creates a
relative minimum for its particular solution.
c)Find the particular solution y=f(x)) to the given differential
equation with
initial condition f(0) = 2
d)For the solution in part c), find lim x aproaches 0
f(x)-2/tan(x^2+2x)

1252) y=(C1)exp(Ax)+(C2)exp(Bx)+F+Gx is the general solution of
the second order linear differential equation:
(y'') + ( -9y') + ( 20y) = ( -9) + ( -8)x. Find A,B,F,G, where
A>B. This exercise may show "+ (-#)" which should be enterered into
the calculator as "-#", and not "+-#". ans:4

1252) y=(C1)exp(Ax)+(C2)exp(Bx)+F+Gx is the general solution of
the second order linear differential equation:
(y'') + ( -4y') + ( 3y) = ( 2) + ( -7)x. Find A,B,F,G, where
A>B.

given the differential equation y'+y=x^2+2x, compute the value of
the solution y(x) at x=1 given the inital condition y(0)=2. Also,
use h=0.1

Differential Equations problem
If y1= e^-x is a solution of the differential equation
y'''-y''+2y=0 . What is the general solution of the differential
equation?

Verify that the function ϕ(t)=c1e^−t+c2e^−2t is a solution of
the linear equation
y′′+3y′+2y=0
for any choice of the constants c1c1 and c2c2. Determine c1c1
and c2c2 so that each of the following initial conditions is
satisfied:
(a) y(0)=−1,y′(0)=4
(b) y(0)=2,y′(0)=0

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