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Math Help - trouble understanding this problem

  1. #1
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    trouble understanding this problem

    A>0 Assuming a function
    f(x)=Sum(0 to infinity) cnxn
    is the solution to the differential equation:

    f''(x)=af(x)

    with the conditions f(0)=1 f'(0)=0 find a formula for the coefficients of f. what is its radius of convergence?

    completely lost and do not understand the problem someone point me in the right direction??
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  2. #2
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    Re: trouble understanding this problem

    Start by writing down your series for f(x), differentiate it twice for f''(x), substitute them into your DE, compare the coefficients of the terms that have like powers of x.
    Thanks from jvang1222
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  3. #3
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    Re: trouble understanding this problem

    Quote Originally Posted by Prove It View Post
    Start by writing down your series for f(x), differentiate it twice for f''(x), substitute them into your DE, compare the coefficients of the terms that have like powers of x.
    thanks very much for your reply

    but I believe i understood up to the "compare the coefficents..." line

    I attempted the second derivative (i believe its right)
    f''(x) = SUMn to infinity n(n-1)cnxn-2

    Substituting this into the equation

    SUMn to infinity n(n-1)cnxn-2 = a(SUMn to infinity cnxn)

    and then i don't understand what you mean by compare the coefficients that have like powers of x

    do you mean that bring a into the sum SUMn to infinity acnxn) in which case a=n(n-1)???

    I also am not sure how to deal with the f(0)=1 and f'(0)=0 conditions

    in the case of f(x)=SUMn to infinity cnxn if x = 0 then the series = 0 meaning i have to add +1 to equation to make the condition correct
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  4. #4
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    Re: trouble understanding this problem

    Quote Originally Posted by jvang1222 View Post
    thanks very much for your reply

    but I believe i understood up to the "compare the coefficents..." line

    I attempted the second derivative (i believe its right)
    f''(x) = SUMn to infinity n(n-1)cnxn-2

    Substituting this into the equation

    SUMn to infinity n(n-1)cnxn-2 = a(SUMn to infinity cnxn)

    and then i don't understand what you mean by compare the coefficients that have like powers of x

    do you mean that bring a into the sum SUMn to infinity acnxn) in which case a=n(n-1)???

    I also am not sure how to deal with the f(0)=1 and f'(0)=0 conditions

    in the case of f(x)=SUMn to infinity cnxn if x = 0 then the series = 0 meaning i have to add +1 to equation to make the condition correct
    f(x) = $c_0+c_1 x + c_2 x^2 \dots$

    f''(x) = $2 c_2 + 6c_3 x+12 c_4 x^2 \dots = a f(x) = a\left(c_0+c_1 x + c_2 x^2 \dots\right)$

    now compare like powers of $x$ and equate their coefficients

    $c_2 = a c_0, 6c_3=a c_1, 12c_4=a c_2$, etc.

    two initial conditions will completely specify a 2nd order linear differential equation so plug in your conditions and solve for the resulting coefficients.
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  5. #5
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    Re: trouble understanding this problem

    Quote Originally Posted by romsek View Post
    f(x) = $c_0+c_1 x + c_2 x^2 \dots$

    f''(x) = $2 c_2 + 6c_3 x+12 c_4 x^2 \dots = a f(x) = a\left(c_0+c_1 x + c_2 x^2 \dots\right)$

    now compare like powers of $x$ and equate their coefficients

    $c_2 = a c_0, 6c_3=a c_1, 12c_4=a c_2$, etc.

    two initial conditions will completely specify a 2nd order linear differential equation so plug in your conditions and solve for the resulting coefficients.
    okay you clarified the cofficents idea but now in terms of deriving an equation for f would it be something like

    c0=2c2/a and then a general equation of cn=2cn+2/a
    the deriving equation line has me confused. am I solving for a?? and then substituing that into the f(x) formula? ex.

    a = 2c_n+2 / c_n and then substituting into f(x)=\sum_{0}{infinity} 2cn+2cn/cnxn

    and upon figuring this all out i have to find the radius of convergence of the formula?!
    Last edited by jvang1222; March 16th 2014 at 04:31 PM.
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  6. #6
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    Re: trouble understanding this problem

    The initial conditions give you c_0 and c_1, and then you figure out what the rest of the c's are using the recurrence relation ( c_{n+2} in terms of c_n).

    Once you have an expression for c_n, you should be able to use the ratio test to get the radius of convergence (it's almost always the ratio test).

    - Hollywood
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