# Question about a Linear 1st Order DE

In summary, the conversation discusses two methods for finding the general solution to the first-order differential equation dy/dx=5y: separation of variables and variation of parameters. The conversation also addresses the placement of the constant in the solution.
[SOLVED] Question about a Linear 1st Order DE

## Homework Statement

Find the general solution to $$\frac{dy}{dx}=5y$$

Now I know that this is separable. But it is in the exercise set that immediately follows "finding a general solution" in which they use variation of parameters. At least I think that is what it is called...when for y'+P(x)y=f(x) you find the integrating factor $\mu(x)=e^{\int P(x)dx}[/tex] Now if I solve by separation: $$\frac{1}{5}\frac{dy}{y}=dx$$ $$1/5\ln y=x+C$$ <----is there a preference of where the C goes (with x or y)?By Var of Parameters: since P=-5, [itex]\mu(x)=e^{-5x}$

$$e^{-5x}*y=C$$

Now is $$1/5\ln y=x+C$$ equivalent to $$e^{-5x}*y=C$$?I mean they must be, but I just can't see it.

Last edited:
I see it! Exponentiate!

## What is a linear 1st order differential equation?

A linear 1st order differential equation is an equation that involves a function and its derivative, where the function and its derivative are both raised to the first power and are not multiplied together. It can be written in the form: y' + p(x)y = g(x), where y is the function, y' is its derivative, p(x) is a function of x, and g(x) is another function of x.

## How do you solve a linear 1st order differential equation?

To solve a linear 1st order differential equation, you can use the method of separation of variables. This involves isolating the dependent variable and its derivative on one side of the equation, and all other terms on the other side. Then, you can integrate both sides and solve for the dependent variable.

## What is the general solution of a linear 1st order differential equation?

The general solution of a linear 1st order differential equation is the set of all possible solutions that satisfy the equation. It includes a constant of integration, which can have different values for different specific solutions.

## What are initial conditions in the context of a linear 1st order differential equation?

Initial conditions refer to the specific values of the dependent variable and its derivative at a given point. These values are used to determine the unique solution to the differential equation.

## Can a linear 1st order differential equation have multiple solutions?

No, a linear 1st order differential equation has a unique solution. This means that given the same initial conditions, there is only one possible solution that satisfies the equation.

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