Question

eliminate the arbitrary constant
y=c1x^2 + c2e^-x
c1= c sub 1
c2= c sub 2
ty

Answer 1

??

Answer 2

anymore info?

Answer 3

set c1 and c2 = some random number?

Answer 4

y=0 is a solution

Answer 5

take the derivatives

Answer 6

y=c1x^2 + c2e^-x
dy/dx = c1*2x - c2*e^(-x)
d^2y/dx^2 = c1*2 + c2*e^(-x)
d^3 y/dx^2 = - c2*e^(-x)

Answer 7

now we can eliminate c1 and c2

Answer 8

y=c1x^2 + c2e^-x
y ' = c1*2x - c2*e^(-x)
y ' ' = c1*2 + c2*e^(-x)
y ' ' ' = - c2*e^(-x)

Answer 9

notice that y ' ' + y ' ' ' = 2*c1

Answer 10

yeah, i got the whole thing, thank you so much master perl

Answer 11

:D

Answer 12

just to confirm, the derivative of 2c1 is 2 and not c1?

Answer 13

no, the derivative of 2*c1 is zero, since it is a constant times constant

Answer 14

i made the same mistake, i thought it represented a variable. but c sub 1 and c sub 2 represent arbitrary constants (which you pick)

Answer 15

a constant times a constant will be a constant, so the derivative of 2 * c_1 is 0

Answer 16

y=c1x^2 + c2e^-x
y ' = c1*2x - c2*e^(-x)
y ' ' = c1*2 + c2*e^(-x)
y ' ' ' = 0 - c2*e^(-x)

Answer 17

wait, woops

Answer 18

yeah, it's so clear to me now, i can handle it from here, again thank you so much master perl

Answer 19

we are looking for an ODE that has no arbitrary constants c_1 , c_2

Answer 20

which satisfies y = C_1*x^2 + C_2 * exp(-x)

Answer 21

y=c1x^2 + c2e^-x
y ' = c1*2x - c2*e^(-x)
y ' ' = c1*2 + c2*e^(-x)
y ' ' ' = - c2*e^(-x)
y '''' = c2*e^(-x).
therefore
y ' ' ' + y ' ' ' ' = 0

Answer 22

and according to wolfram the solution of
y ' ' ' + y ' ' ' ' = 0
is equal to
y(x) = c_4 x^2+c_3 x+c_1 e^(-x)+c_2
Now set c_3 = 0 and c_2 = 0, and we have our solution y given above.

http://www.wolframalpha.com/input/?i=solve+y%27%27%27+%2B+y%27%27%27%27+%3D+0+

Answer 23

@dan815 want to check this makes sense?

Answer 24

how can i thank you more? you're just so awesome, by any chance are you a prof?

Answer 25

again, to be clear what the question is asking
Find a differential equation that y satisfies. The ODE should not contain any arbitrary constants. ANd y = c_1*x^2 + c_2 * e^(-x)

Answer 26

not yet a professor

Answer 27

yeah, been trying to eliminate that for about 4 hrs now and here you are, you saved me, you're gonna be a great prof perl

Answer 28

thanks :D

Answer 29

nice :)

Answer 30

but next midorima, give more information in the question !!! then you might get more replies

Answer 31

next time

Answer 32

sure i will :)

Answer 33

this is a good example which is similar
http://math.stackexchange.com/questions/469049/differential-equation-elimination-of-arbitrary-constants?rq=1

Answer 34

if you have any questions about this one, let me know