Time:
55
Minutes
assignment
MATH. DEPT.
Ordinary Differential Equations I
Math 7
Answer the following questions
QUESTION ONE
a] Find the general solution
y(t ) of the of the equation
ty  = 7t − 6 y
b] Determine (without solving the problem) an interval in which the solution of the given initial value
problem is certain to exist, Justify your answer.
(t 2 − 1) y + 7ty = 2t 3 ,
y ( 0) = 1
QUESTION TWO
a] Show that any separable equation
dy
M ( x) + N ( y )
= 0.
dx
Is also exact.
b] Find the value of b for which the following differential equation is exact, and then
Solve it using the value of b.
( y cos x + bxe y ) + (sin x + x 2 e y − 1)
dy
= 0.
dx
QUESTION THREE
a] Find the longest interval in which the solution of the initial value problem
(t 2 − 3t ) y + ty − (t + 3) y = 0,
y(1) = 2, y(1) = 1
is certain to exist
b] If the functions y1 and y2 are linearly independent solutions of y + p(t ) y + q(t ) y = 0
Prove that y3 = y1 + y2 and y4 = y1 − y2 also form set of linearly independent solutions,
QUESTION FOUR
a] If the functions y1 and y2 are linearly independent solutions of t y − 2 y + (3 + t ) y = 0
2
W ( y1 , y2 )(2) = 3, find the value of W ( y1 , y2 )(4),
b] Find the solution of the following differential equation
y + 2 y + 5 y = 0
And show that the solutions are fundamental