Question

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.

Answer #1

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=

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),...

Solve the following Differential equations
a) x sin y dx + (x^2 + 1) cos y dy = 0

1. If x1(t) and x2(t) are solutions to the differential
equation
x" + bx' + cx = 0
is x = x1 + x2 + c for a constant c always a solution? Is the
function y= t(x1) a solution?
Show the works
2. Write sown a homogeneous second-order linear differential
equation where the system displays a decaying oscillation.

This is a differential equations problem:
use variation of parameters to find the general solution to the
differential equation given that y_1 and y_2 are linearly
independent solutions to the corresponding homogeneous equation for
t>0. ty"-(t+1)y'+y=18t^3 ,y_1=e^t ,y_2=(t+1)
it said the answer to this was C_1e^t + C_2(t+1) -
18t^2(3/2+1/2t)
I don't understand how to get this answer at all

The indicated functions are known linearly independent solutions
of the associated homogeneous differential equation on (0, ∞). Find
the general solution of the given nonhomogeneous equation.
x2y'' + xy' + y = sec(ln(x))
y1 = cos(ln(x)), y2 = sin(ln(x))

Show that the curve
x = 7 cos(t), y = 6 sin(t) cos(t)
has two tangents at (0, 0) and find their equations.

Consider the parametric equations below.
x = t sin(t), y = t
cos(t), 0 ≤ t ≤ π/3
Set up an integral that represents the area of the surface
obtained by rotating the given curve about the x-axis.
Use your calculator to find the surface area correct to four
decimal places

Find an appropriate integrating factor that will convert the
given not exact differential equation cos x d x + ( 1 + 2 y ) sin
x d y = 0 into an exact one. Then solve the new exact
differential equation.

Use variation of parameters to find a general solution to the
differential equation given that the functions y 1 and y 2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
ty"-(t+1)y'+y=30t^2 ; y1=e^t , y2=t+1
The general solution is y(t)= ?

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