Non-homogenous differential equation

James889
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Hi,

My memory ran away, i completely forgot how to do this =/

I need to solve
y'' + 4y = tsint,~~y(0) = 0,~y'(0) = 1

with the general solution C_1sin(2t) + C_2cos(2t)

y = Atsin(t) +Btcos(t)

y' = Asin(t) + Atcos(t) +Bcos(t) -Btcos(t)

y'' = Acos(t) + Acos(t) -Atsin(t) - Bsin(t) -Bsin(t) -Btsin(t)

substitute back into the first equation:

2Acos(t) - 2Bsin(t) + 3Atsin(t) + 3Btcos(t)

So if i let A be 1/3 i get 2/3cos(t) + tsin(t)

How did you determine the C_1 and C_2 constants ?
 
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James889 said:
Hi,

My memory ran away, i completely forgot how to do this =/

I need to solve
y'' + 4y = tsint,~~y(0) = 0,~y'(0) = 1

with the general solution C_1sin(2t) + C_2cos(2t)

y = Atsin(t) +Btcos(t)
Your particular solution needs two more terms:
yp = Asin(t) + Bcos(t) + Ctsin(t) + Dtcos(t)
James889 said:
y' = Asin(t) + Atcos(t) +Bcos(t) -Btcos(t)

y'' = Acos(t) + Acos(t) -Atsin(t) - Bsin(t) -Bsin(t) -Btsin(t)

substitute back into the first equation:

2Acos(t) - 2Bsin(t) + 3Atsin(t) + 3Btcos(t)

So if i let A be 1/3 i get 2/3cos(t) + tsin(t)

How did you determine the C_1 and C_2 constants ?
From the initial conditions.
 
ARGH!

I struggled with this all day trying to figure out what i did wrong..
 
Right

y = ASin(t) + BCos(t) +CtSin(t) + DtCos(t)

y' = ACos(t) - BSin(t) + CSin(t) + CtCos(t) + DCos(t) -DtSin(t)

y'' = -ASin(t) -BCos(t) + CCos(t) +CCos(t) -CtSin(t) - DSin(t) - DSin(t) -DtCos(t)


= -ASin(t) - BCos(t) +2CCos(t) -2DSin(t) -CtSin(t) -DtCos(t)


Adding 4y gives me:

3ASin(t) + 3BCos(t) - 2DSin(t) + 2CCos(t) + 3CtSin(t) + 3DtCos(t)

So C = 1/3
and B = -2/9

But this does not satisfy y(0) = 0
 
This is your particular solution. The general solution is the solution to the homogeneous problem (you miscalled it the general solution) plus the particular solution. Use the general solution and the initial conditions to find C1 and C2.
 
James889:

If you have problem finding the particular solution using Undetermined Coefficients method,
you can always try using the inginiuos Variation of Parameters by Lagrange ;)

y1(x), y2(x) : are two linearly independent homogeneous solutions.
g(x) : the particular expression, in your case = x*sin(x).

u'1(x)*y1(x) + u'2(x)*y2(x) = 0
u'1'(x)*y'1(x) + u'2(x)*y'2(x) = g(x)

solve the system of equations and find u1(x) and u2(x) by integration.
the particular part of the answer will be:

Y(x) = u1(x)*y1(x) + u2(x)*y2(x).

the complete answer for the ODE (A,B are constants):

y(x) = [A*y1(x)] + [B*y1(x)] + [u1(x)*y1(x) + u2(x)*y2(x)].

*Lagrange was a genius !
 
Mark44 said:
This is your particular solution. The general solution is the solution to the homogeneous problem (you miscalled it the general solution) plus the particular solution. Use the general solution and the initial conditions to find C1 and C2.
Hm,
So if i put
y =\frac{t}{3}sin~t - \frac{2}{9}cos~t + C_1sin(2t) + C_2cos(2t)for the condition y(0) = 0 C_2 = \frac{2}{9}

then i have
y' = \frac{t}{3}cos~t + \frac{2}{9}sin~t + 2C_1cos(2t) -2C_2sin(2t)

and for
y'(0) = 1

\frac{1}{3} + 2C_1 = 1, \rightarrow C_1 = \frac{2}{6}

Is this correct?
 
Check it for yourself. You have y = (1/3) tsin(t) - (2/9) cos(t) + (1/3) sin(2t) + (2/9) cos(2t).

This is the solution if
1) it satisfies the initial conditions y(0) = 0 and y'(0) = 1, and
2) it satisfies the diff. equation y'' + 4y = tsin(t).
 
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