Find the solution of the differential equation that satisfies the given initial condition? y(x+1)+y′=0 , y(−2)=1 I don't understand what this question is telling me to do? Can someone please explain?

Salvador Whitehead

Salvador Whitehead

Answered question

2022-11-25

Find the solution of the differential equation that satisfies the given initial condition?
y ( x + 1 ) + y = 0, y ( 2 ) = 1
I don't understand what this question is telling me to do? Can someone please explain?

Answer & Explanation

Nicholas Lara

Nicholas Lara

Beginner2022-11-26Added 10 answers

It is asking you to find the general solution y ( x ) to the differential equation and then to use the initial condition y ( 2 ) = 1 to determine the constant.
Since the equation is separable you can write it in the form d y d x = y ( x + 1 ) and from there 1 y d y = ( x + 1 ) d x you should be able to solve from here by integrating. A constant of integration will be introduced and that is why we have the intial condition y ( 2 ) = 1 to determine this constant.
nazismes2w7

nazismes2w7

Beginner2022-11-27Added 1 answers

This is a linear homogeneous ODE and can be solved using separation.
y ( x + 1 ) + y x = 0 y ( x + 1 ) = y x ( x + 1 ) x = y y , y 0 ( x + 1 ) x = y y , y 0 x 2 2 + x + C = ln ( y ) , y > 0 x 2 2 x + D = ln ( y ) , y > 0 e x 2 2 x + D = y , y > 0 y = A e x 2 2 x , y > 0
now lets use y ( 2 ) = 1 and we get:
1 = A e ( 2 ) 2 2 ( 2 ) = A e 2 + 2 = A e 0 A = 1
hence we get: y = e x 2 2 x , y > 0
In case I've miscalculated, I believe you can understand the general technique of the solution.

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