It is simply the "Separation of variables". The separation enables you to rewrite the differential equations so as to achieve a similarity of measures between two integrals we can assess. A separable equation is actually the first order differential equations that can be straightaway solved using this technique.

A function of two independent variables is said to be separable if it can be demonstrated as a product of 2 functions, each of them based upon only one variable. So, A separable differential equation can be written in the form of \[\frac{dy}{dx} = f(x)g(y) dx dy = f(x)g(y)\].

The process takes place in only 3 easy Steps:

Step 1: Bring all the ‘y’ products (including dy) to one side of the expression and all the ‘x’ terms (including dx) to the other side of the equation.

Step 2: Integrate one side concerning ‘y’ and the other side concerning ‘x’. Don't you forget "+ C" [the constant of integration]

Step 3: Simplify

How to solve separable differential equations is not that difficult as it seems to be, especially, if you have understood the theory of differential equations.

Now you will find detailed solutions to Differential Equations by Variable Separable Method. Based on f(x) and g(y), these mathematical expressions can be solved systematically.

Example1

Solve and identify a general solution to the differential equation.

y ' = 3 e y x 2

Solution

You need to first rewrite the provide equation in form of a differential equation and with variables isolated (separated), the x's on one side while the y's on the other side as given below.

e -y dy = 3 x 2 dx

Integrate both the sides

ò e -y dy = ò 3 x 2 dx

which presents to you

-e -y + C1 = x 3 + C2, C1 and C2 that are constant of integration.

Now, Solve the above equation for y

y = - ln( - x 3 - C ) , where C = C2 - C1.

Then, confirm that the solution acquired fulfills the differential equation given above.

Example2

Find and solve the differential equation using the variable separable e method (x² + 4)y′ = 2xy.

Solution

The product xy on either sides does not permit isolating (separating) the variables. Hence, we make the substitution:

xy=tory=tx.

The connection for differentials is provided by

dy=xdt−td x x².

Substituting this into the expression, we can write in form:

tx (1 + t) dx = x(1−t)x dt − td x x².

Then, By multiplying each of the sides by x and then canceling the correlating fractions in the right and the left, we obtain

t(1+t)dx = (1−t)(xdt−tdx).

Make a note that x=0 is a solution of the equation, which you can verify by direct substitution.

Now, Simplifying the last mentioned equation:

tdx + t²dx = xdt − tdx − xtdt + t²dx,⇒2tdt = x(1−t)dt.

Now the variables x and t have been separated:

2dxx = (1−t)dtt or 2dxx = (1t−1)dt.

After integrating we obtain

2 ∫dxx=∫(1t−1)dt+C,⇒2ln|x|=ln|t|−t+C,⇒lnx²=ln|t|−t+C.

By inducing the reverse substitution t=xy, you discover the general solution of the equation:

lnx²=ln|xy|−xy+C,⇒ln∣∣∣xyx²∣∣∣−xy+C=0,⇒ln∣∣yx∣∣−xy+C=0.

Finally, a complete solution is written in the form:

ln∣∣yx∣∣−xy+C=0,x=0.

By the rule of Separability, a first-order differential equation is called a separable equation, provided after solving it for the derivative,

dy

dx

= F(x, y),

Next, The right-hand side can be factored (divided) as “a formula of just x ” times “a formula of just y ”,

F(x, y) = f [x]g[y]

If you observe that this factoring is not possible, the equation is not separable.

In short, a first-order differential equation is said to be separable if and only if it can be written in the form of:-

dy

dx

= f [x]g[y)] in which f and g are definite functions.

ü separation of variables is also known as the Fourier method within mathematics

ü Crucial logistic differential equation are also separable

ü Newton's Law of Cooling contributed to continuance of separable differential equations

ü partial differential equations variable separable method is used when the partial differential equation and the boundary situations are linear and homogeneous

ü A 'constant of integration' only provides a family of functions that develops a general solution when solving a differential equation.

ü You will always need to add a constant when finding the indefinite integral. Just can ignore it

ü You will be able to find the solutions to specific separable differential equations by separating variables, integrating in regard to 't',and finally solving the resulting algebraic equation for 'y'.

ü Separable differential equation enables us to effectively solve many significant physical occurrences that occur in the world around us. For example, problems of growth and decay.

FAQ (Frequently Asked Questions)

How are Separable Equations Useful?

This division of the differential equation is termed as separable because the variables ‘xx’ and ‘yy’ can be re-positioned to opposite sides of the equation. Then, integrating both sides of the equation provides ‘yy’ as a function of xx, thus solving the differential equation. This property of integration makes Separable differential equations highly useful in computing many physical systems such as:-

ü the rates of chemical reactions

ü the movement of projectiles

ü the growth of populations

Are all Differential Equations Separable?

A first order differential equation y′=f(x,y) is said to be a separable equation, given that the function f(x,y) can be factored (divided) into the product of 2 functions of x and y: f[x,y]=p[x]h[y], where p[x] and h[y] are continuous functions. Moreover, only a first order differential equation is exact if and only if it holds preserved quantity. For instance, separable equations are always exact, in consideration of their explanation that they are of the form: M(y)y + N(t)=0, ... hence, ϕ(t, y) = A(y) + B(t) is a preserved quantity.