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| PAGE: 1180 | SET: Exercises | PROBLEM: 21 |
(a)
The differential equation is 
.
The differential equation is in the form of 
.
is called complementary equation.
The general solution of 
is 
.
The auxiliary equation is 
.
The roots of auxiliary equation is 
.
and .
The roots of auxiliary equation is real and equal.
The solution of complementary equation is 
.
Take 
.
The 
is exponential function and continuous for all values of 
.
The general solution of 
is 
.
.
Substitute 
, 
and 
in 
.
.
Substitute 
in the general solution of .
.
The solution of differential equation is .
Substitute and .
.
(b)
The differential equation is .
Solving non-homogenous differential equation:
If the differential equation is in the form of 
, then general solution of the non-homogenous differential equation is 
, where 
is the general solution of the complementary equation and 
is the particular solution.
General solution of the complementary equation:
If the differential equation is in the form of , then general solution of the complementary equation is 
Particular solution of the differential equation :
If the differential equation is in the form of then the particular solution of the equation is 
, where
and 
.
Here 
is the wronskian of 
and 
.
.
The differential equation is in the form of .
is called complementary equation.
The general solution of is .
The auxiliary equation is .
and .
The roots of auxiliary equation is real and equal.
The solution of complementary equation is .
The general solution of is .
The particular solution of the differential equation is in the form of , where 
and
.
Find wronskian of 
and 
is
Find 
.
Find 
.
Substitute the values of , , 
and in .
General solution of the differential equation is .
Solution of the differential equation is 
.
(a)
(b)
Solution of the differential equation is 
.
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