# Solve the Differential Equation #dy/dx +3y = 0# with #x=0# when #y=1#?

# y = e^(-x) #

We have:

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

The given equation is already in standard form, So

Then the integrating factor is given by;

Which we can directly integrate to get:

Thus,

And the given answer is incorrect.

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The solution is:

Solve the homogeneous differential equation:

The characteristic equation is:

so the general solution of the homogeneous equation is:

Search now a particular solution using Lagrange methods of variable coefficients in the form:

Substituting into the equation:

using the product rule:

The general solution of the equation is then:

In fact:

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After taking Laplace transformation both sides,

After taking inverse Laplace transform, I found

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To solve the differential equation ( \frac{dy}{dx} + 3y = 0 ) with the initial condition ( y(0) = 1 ), we can separate variables and integrate both sides.

Starting with the given differential equation: [ \frac{dy}{dx} + 3y = 0 ]

Separating variables: [ \frac{dy}{y} = -3dx ]

Integrating both sides: [ \int \frac{1}{y} dy = \int -3dx ]

[ \ln|y| = -3x + C ]

where ( C ) is the constant of integration.

Applying the initial condition ( y(0) = 1 ): [ \ln|1| = -3(0) + C ] [ 0 = C ]

Thus, the particular solution is: [ \ln|y| = -3x ]

To find ( y ): [ |y| = e^{-3x} ]

Applying the initial condition again: [ |1| = e^{0} ] [ 1 = 1 ]

So, the solution to the differential equation with the initial condition is: [ y(x) = e^{-3x} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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