Applied Differential Equations (MT2043) Assignment 02
Submission Date: 22/03/2012
Instructor: Dr. Rashid Ali
Q1. Answer the following questions briefly:
a) Explain what is meant by an exact differential equation of order one. How do you
solve such equations?
b) What is the standard form of a linear differential equation of order one? How do
solve such equations?
c) How can one check that if a differential equation of order one is homogenous or not?
How do you solve a homogenous differential equation?
Q2. Solve the equation 0)22()43( 22 =+++ dyyxdxxyx
Q3. Solve the IVP 2)0( ; 0)sin()3cos2( 232 ==−−++ ydyyyxxdxyxyx .
Q4. Determine the constant A to make the following equations exact:
a) 0)4()3( 22 =+++ dyyAxdxxyx
b) 0
11
223 =
⎟
⎠
⎞
⎜
⎝
⎛−+
⎟
⎠
⎞
⎜
⎝
⎛+dy
x
x
dx
x
y
x
Ay ; solve the resulting DE
Q5. Solve the equation 0))cos()sin(())cos()sin(( 22 =+++ dyxyyxxyxdxxyxyxyy
Ans. xysin(xy) = C
Q6. The sine integral function is defined by ∫
=xdt
t
t
xSi 0
sin
)( , where the integrand is defined to
be 1 at t = 0. Express the solution of the IVP
0)1( ; sin102 23 ==+ yxyx
dx
dy
x
in terms of Si(x).
Q7. The following system of differential equations is encountered in the study of a special type
of radioactive series of elements:
,
21
1
yx
dt
dy
x
dt
dx
λλ
λ
−=
−=
,
where λ1 and λ2 are constants. Discuss how to solve the system subject to x(0) = x0, y(0) = y0.
Ans. tt e
xyy
e
x
y21
12
101020
12
10
λλ
λλ
λ
λ
λ
λλ
λ
−−
−
−
−
+
−
=
Q8. Solve the following ODE’s
a) 22 yxyyx +=−
′
b) yxeyye yy ′
+= )2( 3
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