Question

It is easy to check that for any value of "c", the function y = ce^(-2x) + e^(-x) is solution of equation y' + 2y = e^(-x). Find the value of "c" for which the solution satisfies the initial condition
y(-1)= 3.
c=?
It is easy to check that for any value of "c", the function y = ce^(-2x) + e^(-x) is solution of equation y' + 2y = e^(-x). Find the value of "c" for which the solution satisfies the initial condition
y(-1)= 3.
c=?
@Mathematics

Answer 1

Substituting y = ce^(-2x) + e^(-x) in y' + 2y = e^(-x);
we get y' + 2(ce^(-2x) + e^(-x)) = e^(-x) => y' = - 2ce^(-2x) - e^(-x).
Integrating we have; y = ce^(-2x) + e^(-x).
using y(-1) = 3 we get; 3 = ce^(2) + e => c = (3 - e)/e^2.

Answer 2

cool man many thanks

Answer 3

welcome :)

Answer 4

There was no need to do the substitution, they already told you that equation for y was a solution. That last line is all that was needed.

Answer 5

so you just put in the y(-1)=3 in the last equation joe?

Answer 6

well true; thanks

Answer 7

explain joe

Answer 8

yes. if the question had been, "verify that this equation y = (whatever) is a solution to the differential equation, then find c such that ...." then you would need to do that.

Answer 9

so all you needed was the fact that y(-1)=3, and you can solve for c:\[3=ce^{-2(-1)}+e^{-(-1)}\iff 3=ce^2+e\iff3-e=ce^2\iff \frac{3-e}{e^2}=c\]its still important to know how to check you answers to differential equations though, so still keep hrish's solution in mind.

Answer 10

alright thanks, check out my latest post