# Can I use Separation of Variables like this? (3 terms)

• TylerH
In summary, Separation of Variables is a method used to solve certain types of equations, including linear equations and partial differential equations. It can be determined if this method is appropriate for an equation by checking if it is separable, meaning it can be expressed as a product of two functions, each involving only one variable. However, there are limitations to using Separation of Variables, as it may not work for non-separable equations or those with complex or non-constant coefficients. This method can only be used for certain types of differential equations, such as first-order or second-order ordinary differential equations, and can work with any number of terms in the equation as long as it is separable.
TylerH
$$\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$$
$$\mbox{Let }z=T(t)X(x)Y(y)$$
$$T'(t)X(x)Y(y)=T(t)X'(x)Y(y)+T(t)X(x)Y'(y) \Rightarrow$$
$$\frac{T'(t)}{T(t)}=\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)} \Rightarrow$$
$$\frac{T'(t)}{T(t)}=A \wedge \left(\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)}=B \rightarrow \frac{X'(x)}{X(x)}=C \wedge \frac{Y'(y)}{Y(y)}=D \right)$$

... and so on.

Yes, separation of variables like that sometimes works. For example, the 2d wave equation for a vibrating drumhead is solved that way. Remember, when you separate the variables like that it will only work if the PDE happens to have solutions like that. So you can always try. Maybe it will work on a particular problem and maybe it won't.

$$f(x) \frac{\partial z}{\partial x} = g(y) \frac{\partial z}{\partial y}$$
$$z=X(x)Y(y)$$
$$f(x)X'(x)Y(y)=g(y)X(x)Y'(y)$$
$$\frac{f(x)X'(x)}{X(x)}=\frac{g(y)Y'(y)}{Y(y)}$$
$$\frac{f(x)X'(x)}{X(x)}=A \wedge \frac{g(y)Y'(y)}{Y(y)}=B$$
... and so on.

... anyone?

TylerH said:

$$f(x) \frac{\partial z}{\partial x} = g(y) \frac{\partial z}{\partial y}$$

In this case the left side can't depend on y and the right on x so they are both constant. If f and g are not zero then zx=C/f(x) and zy=C/g(y).

Integrating the first gives z(x,y) = F(x) + G(y) where F is the antiderivative of C/f(x).
Differentiating this with respect to y gives G'(y) = C/g(y) so your solution is

z(x,y)=F(x)+G(y) +D

Last edited:
OH! I was going about it the wrong way, but with the right technique, I see.

Can you, please, give an example of a PDE that can't be solved by separation (or the general form of one)?

TylerH said:
OH! I was going about it the wrong way, but with the right technique, I see.

Can you, please, give an example of a PDE that can't be solved by separation (or the general form of one)?

There is no reason to expect most pde's are solvable that way. Most can't be solved analytically in the first place. I don't think there is a general form for such but likely most any random PDE you might write down wouldn't be solvable that way, or any other way except numerically. Here's a simple first order one I just made up:

ux + uy = sin(xy2)

I don't know it can't be solved by separation of variables, but I doubt it. In fact, I don't know whether it can be solved analytically at all.

## 1. Can I use Separation of Variables in any type of equation?

No, Separation of Variables can only be used in certain types of equations, such as linear equations and partial differential equations.

## 2. How do I know if Separation of Variables is the right method to solve my equation?

You can determine if Separation of Variables is appropriate by checking if the equation is separable, meaning it can be expressed as a product of two functions, each involving only one variable.

## 3. Are there any limitations to using Separation of Variables?

Yes, Separation of Variables may not work for equations that are not separable or for equations with a complex or non-constant coefficient.

## 4. Can I use Separation of Variables to solve all differential equations?

No, Separation of Variables can only be used to solve certain types of differential equations, such as first-order or second-order ordinary differential equations.

## 5. Is it necessary to have three terms in the equation to use Separation of Variables?

No, Separation of Variables can be used with any number of terms in the equation as long as it is separable.

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