28
1 Mathematical Physics
1.63 Solve:
d
2 y
dx 2 − 8
dy
dx
= −16y
1.64 Solve:
x
2 dy
dx
+ y(x + 1)x = 9x
2
1.65 Find the general solution of the differential equation:
d
2 y
dx 2 +
dy
dx
− 2y = 2cosh(2x)
[University of Wales, Aberystwyth 2004]
1.66 Solve:
x
dy
dx
− y = x
2
1.67 Find the general solution of the following differential equations and write
down the degree and order of the equation and whether it is homogenous or
in-homogenous.
(a) y
−
2
x
y =
1
x 3
(b) y
+ 5y
+ 4y = 0
[University of Wales, Aberystwyth 2006]
1.68 Find the general solution of the following differential equations:
(a)
dy
dx
+ y = e
−x
(b)
d
2 y
dx 2 + 4y = 2 cos(2x)
[University of Wales, Aberystwyth 2006]
1.69 Find the solution to the differential equation:
dy
dx
+
3
x + 2
y = x + 2
which satisfies y = 2 when x = −1, express your answer in the form y =
f (x).
1.70 (a) Find the solution to the differential equation:
d
2 y
dx 2 − 4
dy
dx
+ 4y = 8x
2
− 4x − 4
which satisfies the conditions y = −2 and
dy
dx
= 0 when x = 0.
(b) Find the general solution to the differential equation:
d
2 y
dx 2 + 4y = sin x