Subject Second Order Linear Equations

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Subject Second Order Linear Equations Powered By Docstoc
					         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
   increases.

    (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