# What is the general solution of the differential equation # dy/dx - 2y + a = 0 #?

# y = 1/2 a +Ce^(2x) #

First write the DE in standard form:

This is a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

This is a standard form of a Differential Equation that can be solved by using an Integrating Factor:

And if we multiply the DE [1] by this Integrating Factor we will have a perfect product differential;

This has converted our DE into a First Order separable DE which we can now just separate the variables to get;

Which we can easily integrate to get:

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Use the separation of variables method.

Integrate both sides:

Multiply both sides by 2:

Use the exponential function on both sides:

The inverses on the left disappear:

Adding an arbitrary constant in the exponent is the same a multiplying by an arbitrary constant:

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The general solution of the differential equation ( \frac{dy}{dx} - 2y + a = 0 ) is ( y = \frac{a}{2} + Ce^{2x} ), where ( C ) is an arbitrary constant.

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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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