Question

Solve the following differential equations

1. cos(t)y' - sin(t)y = t^2

2. y' - 2ty = t

Solve the ODE

3. ty' - y = t^3 e^(3t), for t > 0

Compare the number of solutions of the following three initial value problems for the previous ODE

4. (i) y(1) = 1 (ii) y(0) = 1 (iii) y(0) = 0

Solve the IVP, and find the interval of validity of the solution

5. y' + (cot x)y = 5e^(cos x), y(pi/2) = -4

If you can, please show all steps! I don't understand alot of this.

Answer #1

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

Differential equations
Given that x1(t) = cos t is a solution of (sin t)x′′ − 2(cos
t)x′ − (sin t)x = 0, find a second linearly independent solution of
this equation.

solve the ODE solving for the general solutions
y''+y'-12y=sin(t)e^(3t)

In this problem, x = c1 cos t + c2 sin t is a two-parameter
family of solutions of the second-order DE x'' + x = 0. Find a
solution of the second-order IVP consisting of this differential
equation and the given initial conditions.
x(π/6) = 1 2 , x'(π/6) = 0
x=

Differential Equations. ty'+3y=et /t with t > 0
and initial data y(1) = 2

This is for Differential Equations. I was sick last week and
missed class so I do not understand the process behind solving this
question.
Problem 3 Consider the ODE t 2 y'' + 3 t y' + y =
0.
• Factorize the left hand side of the ODE. Hint: One of the
factors is (D + 2 t -1 I).
• Find the fundamental set of solutions.
• Solve the initial value problem t 2 y'' + 3 t...

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

Solve the given differential equation by undetermined
coefficients.
y'' + 2y' + y = 2 cos x − 2x sin x

Initial value problem : Differential equations:
dx/dt = x + 2y
dy/dt = 2x + y
Initial conditions:
x(0) = 0
y(0) = 2
a) Find the solution to this initial value problem
(yes, I know, the text says that the solutions are
x(t)= e^3t - e^-t and y(x) = e^3t + e^-t
and but I want you to derive these solutions yourself using one
of the methods we studied in chapter 4) Work this part out on paper
to...

Partial differential equations
Solve using the method of characteristics
ut +1/2 ux + 3/2 vx = 0 , u(x,0) =cos(2x)
vt + 3/2 ux + 1/2 vx = 0 , v(x,0) = sin(2x)

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