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scoray [572]
3 years ago
14

Given the second order homogeneous constant coefficient equation y′′−4y′−12y=0 1) the characteristic polynomial ar2+br+c is r^2-

4r-12 . 2) The roots of auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). 4) Given the initial conditions y(0)=1 and y′(0)=−26 find the unique solution to the IVP y= .
Mathematics
1 answer:
Vitek1552 [10]3 years ago
7 0

Answer:

The initial value problem y(x) = 4 e^{-2x} -3 e^{6 x}

Step-by-step explanation:

<u>Step1:</u>-

a) Given second order homogenous constant co-efficient equation

y^{ll} - 4y^{l}-12y=0

Given equation in the operator form is (D^{2} -4D-12)y=0

<u>Step 2</u>:-

b) Let f(D) = (D^{2} -4D-12)y

Then the auxiliary equation is (m^{2} -4m-12)=0

Find the factors of the auxiliary equation is

m^{2} -6m+2m-12=0

m(m-6) + 2(m-6) =0

m+2 =0 and m-6=0

m=-2 and m=6

The roots are real and different

The general solution y = c_{1} e^{-m_{1} x} + c_{2} e^{m_{2} x}

the roots are m_{1} = -2 and m_{2} = 6

The general solution of given differential equation is

y = c_{1} e^{-2x} + c_{2} e^{6 x}

<u>Step 3</u>:-

C) Given initial conditions are y(0) =1 and y1 (0) =-26

The general solution of given differential equation is

y(x) = c_{1} e^{-2x} + c_{2} e^{6 x}   .....(1)

substitute x =0 and y(0) =1

y(0) = c_{1} e^{0} + c_{2} e^{0}

1 = c_{1}  + c_{2}    .........(2)

Differentiating equation (1) with respective to 'x'

y^l(x) = -2c_{1} e^{-2x} + 6c_{2} e^{6 x}

substitute x= o and y1 (0) =-26

-26 = -2c_{1} e^{0} + 6c_{2} e^{0}

-2c_{1} + 6c_{2} = -26 .............(3)

solving (2) and (3)  by using substitution method

substitute   c_{2} =1- c_{1} in equation (3)

-2c_{1} + 6(1-c_{1}) = -26

on simplification , we get

-2c_{1} + 6(1)-6c_{1}) = -26

-8c_{1} = -32

dividing by'8' we get c_{1} =4

substitute c_{1} =4 in equation 1 = c_{1}  + c_{2}

so c_{2} = 1-4 = -3

now substitute c_{1} =4 and c_{2} =-3 in general solution

y(x) = c_{1} e^{-2x} + c_{2} e^{6 x}

y(x) = 4 e^{-2x} -3 e^{6 x}

now the initial value problem

y(x) = 4 e^{-2x} -3 e^{6 x}

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Please help me with the below question.
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By letting

y = \displaystyle \sum_{n=0}^\infty c_n x^{n+r}

we get derivatives

y' = \displaystyle \sum_{n=0}^\infty (n+r) c_n x^{n+r-1}

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a) Substitute these into the differential equation. After a lot of simplification, the equation reduces to

5r(r-1) c_0 x^{r-1} + \displaystyle \sum_{n=1}^\infty \bigg( (n+r+1) c_n + (n + r + 1) (5n + 5r + 1) c_{n+1} \bigg) x^{n+r} = 0

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c_1 = -\dfrac{c_0}5 = -\dfrac15

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\displaystyle \sum_{n=0}^2 c_n x^{n + 4/5} = \boxed{x^{4/5} - \dfrac15 x^{9/5} + \frac1{50} x^{13/5}}

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