Solve the Diffrential equation
d²y/dx² + dy/dx + y = Cos 2x
Include complete solution. [ Complementary Function + Particular Integral ]
Answers
GIVEN :–
• A differential equation –
TO FIND :–
• Solution of differential = ?
SOLUTION :–
• Solution of given differential equation –
• Let find C.F. –
• We should write this as –
• Auxiliary equation –
• Hence , C.F. –
• Now let's find P.I. –
• We know that –
• So that –
• Now Rationalization –
• Again using identity –
• Hence , Solution –
Step-by-step-Explanation
Given:
To find:Solve the Differential equation
Solution:
Solution=C.F.+P.I.
To solve the differential equation first write the auxiliary equation.
It can't be factorize,so applying quadratic formula
Hence,
CF
Now,
Find Particular Integral,
for that assume equation to be non-homogeneous
put the value in
and solve for A and B
sin2x and cos2x are linearly independent,compare the coefficient in LHS and RHS
-4A+2B+A=1
or
-3A+2B=1...eq1
-2A-4B+B=0
or
-2A-3B=0...eq2
Solve eq1 and 2 to find value of A and B
- 6A+4B=2
-6A-9B=0
(+) (+) (-)
_________
13B=2
B=2/13
-2A-3×2/13=0
-2A=6/13
A=-3/13
Thus,
Solution of differential equation is
Thus,
This is the completed Solution of given differential equation.
Hope it helps you.