CBSE Class 12 Mathematics Differential Equation Notes

Definition: An equation involving the independent variable x(say), dependent variable y(say) and the differential coefficient of dependent variable w.r.t independent variable i.e dy/dx, d²y/d²x,…. etc is called differential
equation.

Order of a differential equation is the order of the highest order derivative occurring in the differential equation.

Degree of a differential equation is the degree of highest order derivative occurring in the differential equation when the differential coefficients are made free from radicals, fractions and it is written as a polynomial in differential coefficients.

e.g(d²y/d²x)³+sin(dy/dx) = 0 here order is 2 but this differential equation can’t be written in the form of polynomial in differential coefficient Hence its degree not defined.

Linear and Nonlinear Differential Equations: A differential equation in which the dependent variable and its derivatives occur only in the first degree and are not multiplied together, is called a linear differential equation otherwise it is non-linear.

“Formation of differential Equation” To form a DE from a given equation in x and y containing arbitrary constants (parameters) –

“Initial value problem(IVP) is one in which some initial conditions are given to solve a DE”

1. Differentiate the given equation as many times as the number of arbitrary constants involved in it.
2. Eliminate the arbitrary constant from the equations of y, y’, y’’ etc.

Solution of Differential Equations-
1. Variable separable form
2. Homogenous Equations
3. Linear Differential Equations

VARIABLE SEPARABLE FORM If in the equation, it is possible to get all terms containing x and dx to one side and all the terms containing y and dy to the other, the variables are said to be separable,

Procedure to solve:
Consider the equation dy/dx= X.Y, where X is a function of x only and Y is function of y only.
(i) Put the equation in the form 1/Y ,dy=X.dx.
(ii) Integrating both the sides we get

Homogeneous Differential Equations
A differential equation which can be expressed in the form dy/dx=f(x,y) or dx/dy=g(x,y) where, f (x, y) and g(x, y) are homogenous functions of degree zero is called a homogeneous differential equation

Steps for Solving a Homogeneous Differential Equation

1. Rewrite the differential in homogeneous form
2. Make the substitution y = vx orx = vy where v is a variable.
3. Substitute to rewrite the differential equation in terms of v and x or v and y only
4. Follow the steps for solving separable differential equations.
5. Re-substitute v = y/x or v = x / y in the final solution.

Linear Differential Equation: A first-order linear differential equation can be written in the form dy/dx+PY=Q)

where P and Q are constants or function of x only or dx/dy+Px = Qwhere P and Q are constants or function of
y only.

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