Download Problem Sheet 7. Exact Equations, Integrating Factors and

Problem Sheet 7. Exact Equations, Integrating Factors and Homogenous Equations.
For the tutorials on (April 17th and) 1st May
If you run out of time you can skip the questions marked with a †.
1. Solve the following exact differential equations:
d
(y cos(x)) = e2x , subject to the initial condition y = 23 when x = 0;
dx
d
(b)
(y(x2 − 1)) = xe−x ;
dx
1 dy
y
= 2, subject to the initial condition y = 4 when x = 2.
(c)
−
x dx x2
(a)
2. By finding the integrating factors solve the following differential equations:
2
dy
+
y = 4x;
dx x + 2
dy
(b)
+ 4y = e−3x , subject to the initial condition y = 2 when x = 0;
dx
dy
(c) x3
+ x2 y = 4x3 exp(x2 ).
dx
(a)
3. Solve the following differential equations:
(a)
(b)†
(c)†
y 0 + 3y = e−x
subject to y = 5 at x = 0.
dy
= tan(x)y + 1.
dx
y 0 = cosec(x) − y cot(x).
4. Find the general solution of each of the following homogeneous equations:
y 2
dz
z
x
dy
(a)
= +3 ;
(b)
=
.
dx
x
z
dx
x
5. Find the general solution of each of the following homogeneous equations:
6.
†
7.
†
a)
dy
dx
=
1+
c)
du
dt
=
u
t
+
t
u
Show that the equation
y
x
b)
dz
dx
= −
d) xy
dy
dx
=
z(z + x)
−2
x(x − z)
(y + x)2
dy
y
= + e−y/x is homogeneous, and find its general solution:
dx
x
dy
y
By substituting z = 1/y reduce the equation
+ = y 2 to a linear form, and hence obtain its
dx x
general solution.
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