# General Formula using Miscellaneous Substitution

• asteg123
In summary, the individuals in the conversation are discussing a problem involving a differential equation. They are trying to find the general solution and have attempted to rewrite the equation by letting u=x^3 and using the equation d(y/u)=(udy-ydu)/u^2. However, they are stuck and are seeking help.
asteg123
I'm really stomped with this problem... i can't seem to get the answer...

anyway... here's the problem..

(2(x^3) - (y^3))y'=3(x^2)y

and i need to get the general solution...

SO, here's what i did...

I Let
u=x^3 and
du=3x^2dx

so what happens is

2udy-(y^3)dy= ydu

and that's where i got stuck...

i tried using
d(y/u)=(udy-ydu)/u^2

but i can't seem to get rid of the 2 in 2udy and it would be much of a problem if i did that...

could anyone help me??

Rewrite this diff.eq as:
$$y'=\frac{3\frac{y}{x}}{2-(\frac{y}{x})^{3}}\equiv{F}(\frac{y}{x}$$

I'm sure you have solved such problems previously.

## 1. What is the general formula for using miscellaneous substitution?

The general formula for using miscellaneous substitution is:

X = Y + Z - (A * B)

Where X is the final result, Y and Z are the variables to be substituted, and A and B are the values to be substituted for.

## 2. How do you determine which values to substitute in the general formula?

The values to be substituted in the general formula are determined by the specific problem or equation being solved. Look for clues in the given information and try different combinations until the correct values are found.

## 3. Can the general formula using miscellaneous substitution be applied to all types of equations?

Yes, the general formula using miscellaneous substitution can be applied to a wide range of equations, including linear, quadratic, and exponential equations.

## 4. Are there any limitations to using the general formula with miscellaneous substitution?

The general formula using miscellaneous substitution may not be applicable in some cases where there are multiple variables with conflicting values or when the equation is too complex to be simplified using this method.

## 5. How can I check if my solution using the general formula with miscellaneous substitution is correct?

To check if your solution is correct, plug the substituted values back into the original equation and see if it satisfies the equation. If it does, then your solution is likely correct. You can also use a graphing calculator to graph the equation and see if the substituted values produce the expected graph.

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