# Solving x^2 dy/dx = y-xy with y(-1)=-1

In summary, the conversation is about finding implicit and explicit solutions for the equation x^2(dy/dx)=y-xy, with the initial condition y(-1)=-1. The conversation includes an attempted solution and a discussion about a potential mistake in the integration process. The mistake is eventually identified and resolved.

## Homework Statement

Find implicit and explicit solution for

$$x^2\frac{dy}{dx}=y-xy$$ when y(-1)=-1

So far I have:

$$\frac{(1-x)}{x^2}dx=dy/y$$

$$\Rightarrow x^{-2}(1-x)dx=\ln y+C$$

$$\Rightarrow (x^{-2)-x^{-1})dx=\ln y+C$$

$$\Rightarrow -\frac{1}{x}-\ln x=\ln y+C$$

With the initial values, (-1,-1) are outside of the domain. What gives? My integration looks good to me. What am I missing?

Thanks,
Casey

Actually,

$$\int \frac{du}{u} = \ln |u| + C$$

Assuming everything else is correct, that might be your mistake.

Oh, yes, that looks like it would clear up the problem. Thanks Ben. The integration looked a little to easy to be getting the best of me. It's always the details...

Thanks,
Casey

## 1. What is the given differential equation?

The given differential equation is x^2 dy/dx = y-xy.

## 2. What is the initial condition for this differential equation?

The initial condition for this differential equation is y(-1)=-1.

## 3. How can this differential equation be solved?

This differential equation can be solved using the method of separation of variables.

## 4. What are the steps for solving this differential equation?

The steps for solving this differential equation are:

1. Rearrange the equation to separate the variables on each side.
2. Integrate both sides with respect to their respective variables.
3. Add the constant of integration to one side.
4. Solve for y to get the general solution.
5. Use the initial condition to find the particular solution.

## 5. What is the final solution to this differential equation?

The final solution to this differential equation is y = (1-x)/x, with the initial condition y(-1)=-1 resulting in the particular solution y = (1-x)/x.

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