Math 1B Discussion Section Problems
Rob Bayer
November 15, 2007
1. What is the definition of two functions being linearly independent?
2. For each of the following pairs of functions, determine whether they are linearly independent or de-
pendent:
(a) f(x) = ex, g(x) = e−x
(b) f(x) = ex, g(x) = ex+1
(c) f(x) = xe2x, g(x) = e2x
(d) f(x) = ln(x3), g(x) = ln(xπ)
(e) f(x) = sin(x), g(x) = cos(x)
3. Which of the following second order differential equations are linear? Homogeneous?
(a) exy00 + cos(3x2)y0+ 3y= 0
(b) y00 + 3y0+ 7y= cos x
(c) y00 + 3y0+y2= 0
(d) tan(y00) + cos(x)y0=ex
4. Find the characteristic equation for each of the following ODEs:
(a) y00 +y0−6y= 0
(b) y00 =−y
(c) 4y00 + 3y0−y= 0
5. Find the general solution to y00 + 3y0−18 = 0
6. Solve the initial value problem y00 + 3y0−10y= 0 with y(0) = 1 and y0(0) = 3
7. Solve the boundary value problem y00 =ywith y(0) = 0 and y(2) = 2
8. Most of the time (especially in this class), determining whether two functions are linearly independent
can be done by just staring at them and making an educated guess. However, sometimes things
are more complicated and sometimes you need actual proof that they are linearly independent. One
method for doing this is to compute the Wronskian, W(f, g) = f g0−f0g, as it turns out that two
functions are linearly dependent if and only if W(f, g) = 0.
(a) Use the Wronskian to show that eat and ebt are linearly independent whenever a6=b
(Hint: a function y(x) is equal to 0 if and only if y(x) = 0 for every value of x)
(b) Show that eat and teat are linearly independent for any real number a
(c) Show that cos(at) and sin(at) are linearly independent for any real number a6= 0.