# Initial value differential

Problem: Find the solution to the initial Value problem (differential equations)?

y'=-y+e^-2x
y(0) = 3

Attempt: y' -y = e^(-2x) ----- (1)

Linear equation of first order of form y' + y p(x) = q(x)
p(x)= -1
q(x) = e^(-2x)

Find the integrating factor e^∫p(x) dx = e^∫ - dx = e^-x

Multiply equation (1) by the integrating factor e^-x
y' e^-x - y e^-x = e^(-3x) ----- (2)

The left-hand side of (2) is the derivative of y times the integrating factor or (y e^-x)'
(y e^-x)' = e^(-3x) ----- (3)

Integrate both sides of (3)
y e^-x = ∫ e^(-3x)
y e^-x = (-1/3) e^(-3x) + C
multiply everything by e^x

y = (-1/3) e^(-2x) + C
y(0)=3
3 = (-1/3) e^0 + C
C = 3+1/3 = 10/3

y = (-1/3) e^(-2x) + (10/3) e^x

How does this look?

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Curious3141
Homework Helper
Problem: Find the solution to the initial Value problem (differential equations)?

y'=-y+e^-2x
y(0) = 3

Attempt: y' -y = e^(-2x) ----- (1)

Linear equation of first order of form y' + y p(x) = q(x)
p(x)= -1
q(x) = e^(-2x)

Find the integrating factor e^∫p(x) dx = e^∫ - dx = e^-x

Multiply equation (1) by the integrating factor e^-x
y' e^-x - y e^-x = e^(-3x) ----- (2)

The left-hand side of (2) is the derivative of y times the integrating factor or (y e^-x)'
(y e^-x)' = e^(-3x) ----- (3)

Integrate both sides of (3)
y e^-x = ∫ e^(-3x)
y e^-x = (-1/3) e^(-3x) + C
multiply everything by e^x

y = (-1/3) e^(-2x) + C
y(0)=3
3 = (-1/3) e^0 + C
C = 3+1/3 = 10/3

y = (-1/3) e^(-2x) + (10/3) e^x

How does this look?
The first thing you should do is to substitute your solution back into the d.e. Does it solve the equation?

Is this ##y' = -y + e^{-2x}## or is this ##y' - y = e^{-2x}## your equation?

Because in the problem statement, you gave the former, but you solved the latter. The solution for the latter is correct, but if you made a simple algebraic error in bringing the ##y## over, then the solution is obviously wrong.

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LCKurtz
Homework Helper
Gold Member
Problem: Find the solution to the initial Value problem (differential equations)?

y'=-y+e^-2x
y(0) = 3

Attempt: y' -y = e^(-2x) ----- (1)

Linear equation of first order of form y' + y p(x) = q(x)
p(x)= -1
q(x) = e^(-2x)

Find the integrating factor e^∫p(x) dx = e^∫ - dx = e^-x

Multiply equation (1) by the integrating factor e^-x
y' e^-x - y e^-x = e^(-3x) ----- (2)

The left-hand side of (2) is the derivative of y times the integrating factor or (y e^-x)'
(y e^-x)' = e^(-3x) ----- (3)

Integrate both sides of (3)
y e^-x = ∫ e^(-3x)
y e^-x = (-1/3) e^(-3x) + C
multiply everything by e^x

y = (-1/3) e^(-2x) + C
Isn't C included in the "everything"?

Curious3141
Homework Helper
Isn't C included in the "everything"?
Yes, it should be. The reason it didn't affect the answer is because ##e^0 = 1##.

vanhees71
Gold Member
2019 Award
Why not using the standard way of solving linear equations, i.e., first solving the homogeneous equation
$$y'=-y$$
and then using the ansatz of the variation of the constant?

Curious3141
Homework Helper
Why not using the standard way of solving linear equations, i.e., first solving the homogeneous equation
$$y'=-y$$
and then using the ansatz of the variation of the constant?
The integrating factor is the usual elementary method that's taught for questions like this, I think.

I was about to suggest using a Laplace transform, which gives a quick algebraic solution in an initial value problem like this, but thought better of it, simply because it's unlikely the thread starter has covered it yet.

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