1. State whether each of the sets of functions of x below is a linearly independ

Question

1. State whether each of the sets of functions of x below is a linearly independent set of functions: a) sin(x), cos(x), sin(2x), cos(2x), sin(x)oosx) b) x,x(1-?), ?(1-?), ?(1-X) d) e'. e. cosh(x), cosh(2x) e) sin'(x), cos(x), sin(2x), cos(2x) 2. Determine the general solution to the ODE 0.2? + (4901)a 0 Determine the limit of wt) as the time approaches infinity 3. Classify each of the three ODE's as to order, linear/nonlinear, homogeneous/non-homogeneous, and constant, non-constant coefficients: b) ü + a(u2-1) + 311-0 (? given const) i + 100u2e 4. Solve the following initial value problem: u0.16u0 with u(0) 05. Apply the initial conditions to determine the numerical values of constants appearing in the general solution. 5. A homogeneous linear ODE of sixth order with constant coefficients has the following six eigenvalues: Write out the general solution to this ODE model (you should have six undetermined constants). Is it true that u(t) >0 asf for any and all initial conditions? For some initial conditions? Find the homogeneous and the particular solutions to the following non-homogeneous, linear ODE's (problems 6 through 9) + 914 = 2cos3t 10. Determine the particular solution to the following linear, non-homogeneous ODE

Explanation / Answer

I am solving the Q1 with multiple-subparts as per Chegg guidelines

Q1)

a)

sin(x),cos(x),sin(2x),cos(2x), sinxcosx

The function is not the set of linearly independent functions, since

sin(2x) = 2sin(x)cos(x)

So the function sin(2x) and sin(x)cos(x) are independent of each other

b)

The functions are independent, since the highest powers of the function are different, hence none of them depends on each other

c)

The functions are linearly independent, there are two terms which have same power x^2 but (1+x)^2, but due to presence of 1 there are linearly independent

d)

Linearly independent

e)

The functions are not linearly independent

since cos(2x) = cos^2x - sin^2x, which can be written with the function cos^2x and sin^2x

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