1000 Solved Problems in Modern Physics

28 1 Mathematical Physics

1.63 Solve:

d^2 y

dx^2

− 8

dy

dx

=− 16 y

1.64 Solve:

x^2

dy

dx

+y(x+1)x= 9 x^2

1.65 Find the general solution of the differential equation:

d^2 y

dx^2

+

dy

dx

− 2 y=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 xy=x^13
(b)y′′+ 5 y′+ 4 y= 0
[University of Wales, Aberystwyth 2006]

1.68 Find the general solution of the following differential equations:

(a)ddxy+y=e−x

(b)d

(^2) y
dx^2 +^4 y=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 satisfiesy=2 whenx =−1, express your answer in the formy=

f(x).

1.70 (a) Find the solution to the differential equation:

d^2 y

dx^2

− 4

dy

dx

+ 4 y= 8 x^2 − 4 x− 4

which satisfies the conditionsy=−2 andddyx=0 whenx=0.

(b) Find the general solution to the differential equation:

d^2 y

dx^2

+ 4 y=sinx