# Variable Separable

## Some differential equations can be solved by the method of separation of variables. (dy/dx) = f(x, y) is a first order and first degree differential equation. If f(x, y) can be written as h(x).g(y), where h(x) is a function of x and g(y) is a function of y, then we can separate h(x) and g(y). Solutions of differential equations of the type (dy/dx) = f(ax + by + c): Step1: Substitute ax + by + c = v in the given differential equation and change the differential coefficients accordingly.Step2: Obtain the relation f(v)dv = g(x)dxStep3: Integrate the obtained equation to get a relation between x, v and arbitrary constants.Step4: Substitute the value of v in the relation obtained in step-3 to get a relation between x, y and arbitrary constants.Keywords: Variable Separable method, Solutions of differential equations

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• Q1

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• Q2

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• Q3

The general solution of the differential equation

(1 + x2) dy = (1 + y2) dx is

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tan-1y = tan-1x + C.

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• Q4

The general solution of the differential equation y(x2y + ex)dx – exdy = 0 is

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x3y + 3ex = Cy

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Hence, the solution is

$v·{e}^{x}=-\int {x}^{2}{e}^{-x}·{e}^{x}dx+\frac{C}{3}$

• Q5

The differential equation, (ex + 1)ydy = (y + 1)exdx has the solution

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