Question

*y*^{′} + *y* = 5 sin(2*t*)

Answer #1

here we are finding the general solution of y(t)

Eliminate the parameter to find a Cartesian equation of the
curves:
a) x=3t+2,y=2t−3
b) x=21sin(t)−3,y=2cos(t)+5

The homogeneous solutions to an ODE are sin(2t) and cos(2t).
Suppose that the forcing function is 1.5 cos(2t) what is an
appropriate form of the general solution?
y(t)=Acos2t +Bsin2t + C t cos(2t+ᶲ)
, (b) y(t)=Acos2t +Bsin2t
+ C cos2t + Dsin2t
y(t)=Acos2t
+Bsin2t,
(d) y(t)=Acos2t +Bsin2t + C cos(2t+ᶲ)
What is the total number of linearly independent solutions that
the following ODE must have?
y" +5y'+6xy=sinx
Two (b) Four
(c) Three
(d) Five

1.Show that cos 2t, sin 2t, and e^5t are linearly independent
and form a fundamental set of solutions for the equation: y ′′′ −
5y ′′ + 4y ′ − 20y = 0
2.Find the general solution to the equation: y ′′′ − y ′′ − 4y ′
+ 4y = 0

Consider the undamped spring equation
y'' + cy = sin(2t).
(a) For what value of c does resonance occur? Compute the
solution at resonance with y(0) = 1 and y' (0) = 0.
(b) For what values of c is there a beat with frequency 0.1 Hz?
(The beat frequency is defined as (|µ-w|)/2 where µ is the natural
frequency of the spring and ! is the forcing frequency.)

If u(t) = sin(6t), cos(2t), t and v(t) = t, cos(2t), sin(6t) ,
use Formula 4 of this theorem to find d dt u(t) · v(t) .

Differential Equations
Solve for the IVP
( y2 - 2 sin (2t) )dt + ( 1 + 2ty)dy = 0
y (0)= 1

Find the Laplace Transform of the following functions:
1. e^(-2t+1)
2. cos^2(2t)
3. sin^2(3t)

Find a particular solution to ?″+25?=−20sin(5?). y ″ + 25 y = −
20 sin ( 5 t ) . ??=

y'' + 2y' = 4sin(2t)

Find the length of the curve
1) x=2sin t+2t, y=2cos t, 0≤t≤pi
2) x=6 cos t, y=6 sin t, 0≤t≤pi
3) x=7sin t- 7t cos t, y=7cos t+ 7 t sin t, 0≤t≤pi/4

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