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)dx
Step3: 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 step3 to get a relation between x, y and arbitrary constants.
Keywords: Variable Separable method, Solutions of differential equations
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Q1Marks:1
Answer:
Explanation:

Q2Marks:1
Answer:
Explanation:

Q3
The general solution of the differential equation
(1 + x^{2}) dy = (1 + y^{2}) dx is
Marks:1Answer:
tan^{1}y = tan^{1}x + C.
Explanation:

Q4
The general solution of the differential equation y(x^{2}y + e^{x})dx – e^{x}dy = 0 is
Marks:1Answer:
x^{3}y + 3e^{x} = Cy
Explanation:
$\mathrm{We}\mathrm{have},y\left({x}^{2}y+{e}^{x}\right)dx{e}^{x}dy=0$
$\Rightarrow {e}^{x}\frac{dy}{dx}={x}^{2}{y}^{2}+y{e}^{x}$
$\mathrm{Dividing}\mathrm{by}{y}^{2}{e}^{x},\mathrm{we}\mathrm{get}$
$\frac{1}{{y}^{2}}\frac{dy}{dx}\frac{1}{y}={x}^{2}{e}^{x}$
$\mathrm{Putting}\frac{1}{y}=v,\mathrm{we}\mathrm{get}$
$\frac{1}{{y}^{2}}\frac{dy}{dx}=\frac{dv}{dx}$
$\mathrm{Thus},\mathrm{we}\mathrm{have}\frac{dy}{dx}+v={x}^{2}{e}^{x}\mathrm{which}\mathrm{is}\mathrm{a}\mathrm{linear}\mathrm{differential}\mathrm{equation}.$
$\mathrm{I}.\mathrm{F}.={e}^{\int 1dx}={e}^{x}$
Hence, the solution is
$v\xb7{e}^{x}=\int {x}^{2}{e}^{x}\xb7{e}^{x}dx+\frac{C}{3}$
$\mathrm{or}\mathit{}\mathit{}\frac{1}{y}{e}^{x}=\frac{{x}^{3}}{3}+\frac{C}{3}\Rightarrow {x}^{3}y+3{e}^{x}=Cy$

Q5
The differential equation, (e^{x} + 1)ydy = (y + 1)e^{x}dx has the solution
Marks:1Answer:
(1 + y)(e^{x} + 1) = Ce^{y}.
Explanation: