MAT 201E DIFFERENTIAL EQUATIONS WORKSHEET II
Subject: Second Order Linear Equations
1. Find the solution of the following initial value problems and describe its behavior as t
(a) 6y − 5y + y = 0, y(0) = 4, y (0) = 0
(b) y + 3y = 0, y(0) = −2, y (0) = 3
2. Find a diﬀerential equation whose general solution is y = c1 e−t/2 + c2 e−2t .
3. Consider the initial value problem
2y + 3y − 2y = 0, y(0) = 1, y (0) = −β,
where β > 0.
(a) Solve the initial value problem.
(b) Find the coordinates (t0 , y0 ) of the minimum point of the solution for β = 1.
(c) Find the smallest value of β for which the solution has no minimum point.
4. Find the general solution of ty + y = 1, t > 0.
(Hint: For a second order diﬀerential equation of the form y = f (t, y ), the substitution
v = y , v = y leads to ﬁrst order equation of the form v = f (t, v)).
5. Determine the largest interval in which the following initial value problems are certain to
have a unique diﬀerentiable solution. Do not attempt to ﬁnd the solution.
(a) t(t − 4)y + 3ty + 4y = 2, y(3) = 0, y (3) = −1
(b) (x − 3)y + xy + (ln |x|)y = 0, y(1) = 0, y (1) = 1
6. If the Wronskian W of f and g is 3e4t , and if f (t) = e2t , ﬁnd g(t).
7. If f, g and h are diﬀerentiable functions, show that W (f g, f h) = f 2 W (g, h).
8. If y1 and y2 are linearly independent solutions of t2 y − 2y + (3 + t)y = 0 and if
W (y1 , y2 )(2) = 3, ﬁnd the value of W (y1 , y2 )(4).
9. Solve y − 6y + 25y = 0.
10. Solve y − 8y + 16y = 0.
11. Use the method of reduction of order to ﬁnd a second solution of the
(x − 1)y − xy + y = 0, x > 1; y1 (x) = ex .
12. Find a general solution of the given diﬀerential equation.
(a) 2y + 3y + y = t2 + 3 sin t
(b) y − y − 2y = cosh(2t) (Hint: cosh t = (et + e−t )/2)
(c) y + y = 3 sin(2t) + t cos(2t)
13. Determine the form of a particular solution to y + 4y = x2 sin(2x).
14. Solve y + 6y + 9y = e−3x /x5 by using the method of variation of parameters.
15. Show that the given functions y1 and y2 satisfy the corresponding homogeneous equation;
then ﬁnd a particular solution of the given nonhomogeneous equation.
(a) (1 − t)y + ty − y = 2(t − 1)2 e−t , 0 < t < 1; y1 (t) = et , y2 (t) = t
(b) x2 y − 3xy + 4y = x2 ln x, x > 0; y1 (x) = x2 , y2 (x) = x2 ln x