UNIVERSITY OF RUHUNA
DEPARTMENT OF MATHEMATICS
Bachelor of Science (GENERAL) Degree (Level I)
MATHEMATICS
MAT112δ : Differential Equations
Tutorial No:03
Semester I, 2022
Submit answers for question 1(ii), (iv), 2(iv), 3(i), (iii) and 4 on or before
22/12/2022
1. Find the integration factor and then solve the given differential equations.
(i)
dy
dx
− 5y = ex
dy
+ 3y cos (x) = cosec(x)
(ii) sin x dx
′
(iii) y + 4y = e−x
dy
(iv) (x − 1)3 dx
+ 4(x − 1)2 y = x + 1
dy
(v) x dx
= x2 + 3y
2. Determine whether the following differential equations are exact or not?
(i) (x3 + y 3 )dx + 3x2 ydy = 0
(ii) xy 4 dx + 4x2 y 3 dy = 0
(iii) (3x2 + y 2 )dx + (2xy)dy = 0
(iv) cos (y)dx + (y 2 − x sin (y))dy = 0
3. Determine whether the following differential equations are exact. If it exact, solve.
(i) (sin y − y sin x)dx + (cos x + x cos y − y)dy = 0
(ii) (3x2 y + 2)dx + (x3 + y)dy = 0
(iii) (2x + y)dx + (x + 6y)dy = 0
(iv) (y cos x + 2xey )dx + (sin x + x2 ey − 1)dy = 0
4. Solve the IVP,
(3x2 y − 1)dx + (x3 + 6y − y 2 )dy = 0, when y(0) = 3.
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