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T1Solutions.pdf

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Department
Mathematics
Course
MATH 1005
Professor
Ida Toivonen
Semester
Fall

Description
MATH 1005B Test 1 Solutions January 31, 2012 [Marks] [6] 1. Solve the initial-value problem 2y =os(x),y (0) = 2. y Solution: ▯ 2 2yy =cos( x) ⇒ y =sin( x)+ c ⇒ y = ± sin(x)+ c. y(0) = 2 ⇒ 2= √ c ⇒ c =4 ⇒ y = sin(x)+4. 2 2 ▯ x + y [6] 2. Find the general solution of y xy . Soluti2n: 2 ▯ x + y x y y ▯ 1 ▯ 1 ▯ 1 y = xy = y + x,u = x ⇒ u + xu = u + u ⇒ xu = u ⇒ uu = x ⇒ 1u =ln |x| + c ⇒ u = ± 2ln |x| + k ⇒ y = ±x 2ln |x| + k. 2 [6] 3. Find the general solution of xy3 y = x +1. Solution: R y + 3y = x +1 ⇒ I(x)= e xdx = e3ln |= |x| = ±x ,andw emaytek I(x)= x . x x x4 x3 Then x 3y +3 x y = x + x 2 ⇒ (x y) = x + x 2 ⇒ x y = + + c ⇒
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