Solve the differential equation by using method of

Amya Horn

Amya Horn

Answered question

2022-03-31

Solve the differential equation by using method of variation of parametrs
y+9y=1f(x)
take f(x)=cos11x

Answer & Explanation

slokningol09

slokningol09

Beginner2022-04-01Added 6 answers

Given equation is
y+9y=1cos(11x) (i)
Let us consider y=z Then equation (i) is reduced to
z+9z=sec(11x) (ii)
Equation (ii) is a second order linear non-homogeneous differential equation.
Auxiliary equation corresponding to equation (i) is
m2+9=0
m=±3i
So, complementary function is
zc=acos(3x)bsin(3x)
with arbitrary constants a and b.
Let, a particular solution of equation (ii) be
zc=v1cos(3x)+v2sin(3x)
Now,
W(cos(3x),sin(3x))=[cos(3x)sin(3x)3sin(3x)3cos(3x)]=3
Then, v1=sec(11x)sin(3x)W(cos(3x)sin(3x))dx
=13sec(11x)sin(3x)dx
v2=sec(11x)cos(3x)3dx
Therefore, particular solution is
zc=cos(3x)3p(x)+sin(3x)3q(x)
p(x)=sec(11x)sin(3x)dx
q(x)=sec(11x)cosxdx
Then, solution of (ii) is
z=acos(3x)+bsin(3x)+cos(3x)3p(x)+sin(3x)3q(x)
dydx=acos(3x)+bsin(3x)cos(3x)3p(x)+sin(3x)3q(x)
On integration,y=a3sin(3x)+b3cos(3x)13p(x)cos(3x)dx+13q(x)sin(3x)dx+c,
c=integrating constant
So, the general solution of the given equation isa3sin(3x)+b3cos(3x)13p(x)cos(3x)dx+13q(x)sin(3x)dx+c
p(x)=sec(11x)sin(3x)dx
q(x)=sec(11x)cosxdx
It can be noted that due to complexity of the integral, p(x) and q(x) can't be evaluated.
Jeffrey Jordon

Jeffrey Jordon

Expert2022-08-23Added 2605 answers

Answer is given below (on video)

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