Separation of Variables
Some differential equations can be solved by separation of variables
What is separation of variables?
|All the y variables should be moved to one side of the equation, and all the x variables should be moved to the other side.|
- Separate the variables
- Integrate both sides
- Solve for y
- Substitute the initial condition to solve for C
However, some of differential equations are too complicated that we cannot solve them by sepration of variables. Luckily, there is another way to solve these questions.
What is homogeneous differential equation?
Some differential equations that are not separable in x and y can be made separable by a change of variables. This is true for differential equations of the form y'=f(x,y), ia homogeneous of degree n if
|Homogeneous means we can take a function:||f(x,y)|
|Multiply each variable by t:||f(tx,ty)|
|Then we can rearrange it to get this:||tnf(x,y)|
So to be homogeneous a function must pass that test!
In other words: when each variable is multiplied by z, the result is mutiplied by some power of z:
f(tx,ty) = tnf(x,y)
How to varify homogeneous equations?
|start with||f(x,y) = x + 3y|
|Multiply each variable by z:||f(zx,zy) = zx + 3zy|
|Let's rearrange it by factoring out z:||f(zx,zy) = z(x + 3y)|
|And x + 3y is f(x,y):||f(zx,zy) = zf(x,y)|
|Which is what we wanted, with n=1:||f(zx,zy) = z1f(x,y)|
How to solve homogeneous differential equations?
We can solve it using Separation of Variables but first we create a new variable v=y⁄xv = y⁄x is also y = vx And dy⁄dx= d(vx)⁄dx=vdx⁄dx+xdv⁄dx(by the Product Rule) Which can be simplified to dy⁄dx=v+xdv⁄dx
using y=vx and dy⁄dx=v+xdv⁄dx we can solve the Differential Equation.