National University of Computer & Emerging Sciences– FAST
MT203-Differential Equations Computer Science Major
Date: Fri 12th Feb 2010 Due Date: Wed 17th Feb 2010
Tutorial 2
Exact, Integrating Factor, Linear and Bernoulli Differential Equations
Q1. For the given solution u(x y), Find Exact Differential Equation du=0
2
x
(i) y
u=e
u=tan( y ¿−x )¿
2 3
(ii)
1
(iii) u= 2 2
(x + y )
Q2. Show that the following equations are exact and solve them.
(i) − y x−2 dx+ x −1 dy=0
(ii) ( cot y + x 2 ) dx=x cosec 2 ydy
(iii) sinh x cos y dx=cosh x sin y dy
Q3. Solve the initial values problems.
(i) ¿ −¿ e y ¿ dx−x e y dy=0; y ( 1 )=0
π
(ii) 2 sin ωy dx +ω cos ωy dy=0 ; y ( 0 ) =
2ω
(iii) 2 xydy−( x 2 + y 2 ) dx=0 ; y ( 1 )=2
Q4. Show that the given function is an integrating factor and solve.
(i) ydx +¿
(ii)( a+ 1 ) ydx + ( b +1 ) xdy =0 ; F=x a y b
(iii) 2 cos y dx−tan 2 x sin y dy=0 ; F=cos 2 x
Q5. Find an integrating factor.
(i) 2 xydx+ 3 x 2 dy=0 (ii) 2 cos y dx=sin y d y
(iii) 2 x tan y dx + sec 2 ydy =0 (iv)( 2 cos y + 4 x 2 ) dx−x sin y dy=0
Q6. Find the general solutions of the following linear differential equations.
(i) y ' + y sin x=e cos x (ii) y ' + y=e− x tan x
3
'
(iii) y = y tan x ; y ( π )=2 (iv) y ' +6 x 2 y=e−2 x / x 2 ; y (1 )=0
Q7. Solve the Non linear Differential equations.
(i) y ' +2 y= y 2 (ii) y ' + y=−x / y
' 1 1 4
(iii) y ' + xy=x y−1 (iv) y + y= (1−2 x) y
3 3
' tan y ' 1
(v) y = (vi) y = y
( x−1) (6 e −2 x )
