# Solving 3xy^3 + (1+3x^2y^2)dy/dx=0

• jenettezone
In summary, the conversation is about solving the differential equation 3xy^3 + (1+3x^2y^2)dy/dx=0 using integrating factors. The equation is not linear and therefore not separable. The solution involves a substitution of u = xy to make the equation separable.
jenettezone

## Homework Statement

Solve 3xy^3 + (1+3x^2y^2)dy/dx=0 using integrating factors

y' + p(x) = q(x)

## The Attempt at a Solution

I'm having trouble putting the equation to y' + p(x) = q(x)
I distributed dy/dx so it becomes 3xy^3dy/dx + 1dy/dx+3x^2y^2dy/dx=0
But I didn't know where to go from there.
So I multiplied both sides by dx and 3xy^3dx + (1+3x^2y^2)dy=0

For starters, it's y' + P(x)*y = q(x)

For this to be true, the DE has to be linear.

Do you think it is linear, separable or neither?

the definition i have for a linear DE is that it is a DE that can be written in the form y' + P(x)*y = q(x). I am trying to rewrite the DE in that form, but it looks like I can't. If I can't, then according to the definition I have, the equation is not linear, and therefore not separable. But there is an answer from the book's answer set, so it looks like it should be linear...

You won't need to rely upon integrating factors in this case.

we know dy/dx = -3xy^3/(1 + 3x^2y^2)

Thus: dx/dy = -1/3xy^3 - x/y

Making a simple substitution of u = xy

dx/dy = (y*du/dy - u)/y^2 when the substitution is made

The equation should become separable.

ohhh, i see it now. thank you!

Why did you say "the equation is not linear, and therefore not separable"? Most separable equations are not linear. An easy example is dy/dx= x/y.

## 1. What is the purpose of solving this equation?

The purpose of solving this equation is to find the values of x and y that satisfy the equation and make it true. This can help us understand the relationships between the variables and potentially make predictions or solve real-world problems.

## 2. How do I solve this equation step-by-step?

To solve this equation, we can use the following steps:
1. Isolate the term with dy/dx by moving all other terms to the other side of the equation.
2. Factor out the term with dy/dx.
3. Divide both sides by the remaining term with dy/dx.
4. Integrate both sides with respect to x.
5. Solve for y by simplifying the equation.
6. Substitute the values of x and y back into the original equation to check for correctness.

## 3. What should I do if I encounter a term with dy/dx?

If you encounter a term with dy/dx, you should isolate it on one side of the equation and use integration to solve for the remaining variables. Remember to use the chain rule when integrating a term with y raised to a power.

## 4. Are there any special cases I should be aware of when solving this equation?

Yes, there are a few special cases to consider when solving this equation:
- If the equation has multiple terms with dy/dx, you may need to use partial fraction decomposition to solve for y.
- If the equation contains trigonometric functions, you may need to use trigonometric identities to simplify and solve for y.
- If the equation has an initial condition, you can use it to find the constant of integration and get the specific solution.

## 5. How can I check if my solution is correct?

To check if your solution is correct, you can substitute the values of x and y back into the original equation. If the equation holds true, then your solution is correct. You can also graph the equation and your solution to visually check for correctness.

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