By going through these CBSE Class 12 Maths Notes Chapter 9 Differential Equations, students can recall all the concepts quickly.

## Differential Equations Notes Class 12 Maths Chapter 9

**Differential Equations:**

An equation involving the derivative (derivatives) of the dependent variable with respect to the independent variable (variables) is called a differential equation.

e.g. x\(\frac{d y}{d x}\) + y = 0, 2 \(\frac{d^{2} y}{d x^{2}}\) + y^{3} = 0 are the differential equation

**Ordinary Differential Equation:**

A differential equation involving derivatives of the dependent variables with respect to only one independent variable is said to be an ordinary differential equation.

**Partial Differential Equation:**

A differential equation involving derivatives with respect to more than one independent variable is known as a partial differential equation.

Notation: Derivatives may also be written as

\(\frac{d y}{d x}\) = y’, \(\frac{d^{2} y}{d x^{2}}\) = y”, \(\frac{d^{3} y}{d x^{3}}\) = y”’,…………

**Order of Differential Equation:**

The order of the highest order derivative of the dependent variable with respect to independent variable involved in the differential equation is called the order of the differential equation e.g.

\(\frac{d y}{d x}\) and \(\frac{d^{2} y}{d x^{2}}\) + \(\frac{d y}{d x}\)+ y = k involve derivatives whose highest orders are 1 and 2 respectively.

∴ \(\frac{d y}{d x}\) + y = c is of order 1 and \(\frac{d^{2} y}{d x^{2}}\) + \(\frac{d y}{d x}\) + c y = x is of order 2.

∴ \(\frac{d y}{d x}\) + y = c is order 1 an \(\frac{d^{2} y}{d x^{2}}\) + \(\frac{d y}{d x}\) + c y = x is of order 2.

**Degree of differential equation:**

When a differential equation is a polynomial equation in derivatives, the highest power (positive integral index) of the highest order derivative is known as the degree of the differential equation e.g.

1. In \(\left(\frac{d y}{d x}\right)^{2}\) + \(\frac{d y}{d x}\) + y = c, the highest order derivative is \(\frac{d y}{d x}\), its positive integral power is 2. Its degree is 2.

2. In \(\frac{d^{3} y}{d x^{3}}\) + \(\frac{d^{2} y}{d x^{2}}\) + \(\frac{d y}{d x}\) + 4 = 0, the highest order derivative is \(\frac{d^{3} y}{d x^{3}}\). Its positive integral power is 1.

∴ Its degree is one.

**Solution of differential equation:**

The solution of a differential equation is a function y = f(x) which satisfies the given differential equation. It is known as its solution.

**General solution (primitive):**

The solution which contains as many arbitrary constants as the order of the differential equation is said to be the general solution (primitive) of the differential equation.

**Particular solution:**

The solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

**Formation of differential equation:**

To form a differential equation, we proceed as follows:

- Write the given equation.
- Differentiate w.r.t. x successively as many times as the number of arbitrary constants involved in the given equation.
- Eliminate the arbitrary constants.

The resulting equation is the required differential equation e.g.

(a) The equation y = mx has one arbitrary constant m.

∴ Differentiating, we get \(\frac{d y}{d x}\) = m.

Eliminating m, y = \(\left(\frac{d y}{d x}\right)\)x is the required differential equation.

(b) Consider the family of curves

y = a cos (x + b) …………(1)

Here, a and b are the arbitrary constants.

Now, \(\frac{d y}{d x}\) = – a sin (x + b) …….(2)

and \(\frac{d^{2} y}{d x^{2}}\) = – a cos (x + b) ……(3)

Eliminating a and b using (1) and (3), we get

\(\frac{d^{2} y}{d x^{2}}\) = – y

or

\(\frac{d^{2} y}{d x^{2}}\) + y = 0, which is the required differential equation.

**Methods of Solving First Order, First Degree Differential Equations:**

1. Variables are Separable

When the equation may be expressed as \(\frac{d y}{d x}\) = h(y) g(x), then we can write it as \(\frac{d y}{h(y)}\) = g(x) dx.

Intergrating, we get the solution as ∫\(\frac{d y}{h(y)}\) = ∫g(x) dx + C.

2. Homogeneous Differential Equation

Let us write the differential equation as \(\frac{d y}{d x}\) = f(x, y).

Replacing x by λ, x, and y by λy, we get f(λx, λy)= λn f(x, y).

Then the differential equation is homogeneous of degree n.

To solve such an equation

\(\frac{d y}{d x}\) = f(x,y),put y = v x

or

\(\frac{d y}{d x}\) = v + x\(\frac{d y}{d x}\)

We get v + x \(\frac{d v}{d x}\) = f(v)

or

x\(\frac{d v}{d x}\) = f(v) – v

∴ Solution is ∫\(\frac{d v}{f(v)-v}\) = ∫\(\frac{d x}{x}\) + C.

3. Linear Differential Equation

(a) The linear differential equation is of the form \(\frac{d y}{d x}\) + Py = Q, where P and Q are the functions of x.

This is a first-order, first-degree differential equation. To solve the equation, we find the integrating factor

I.F. = e^{∫p dx}.

Then, the solution is

ye^{∫p dx} = ∫Qe^{∫p dx} dx + C.

(b) When the equation is of the form \(\frac{d x}{d y}\) + Px = Q,

where P and Q are the functions of y, then

I.F. = e^{∫p dy}

∴ Solution is

x e^{∫p dy} = ∫Qe^{∫p dy} dy + C.

DEFFRENTIAL EQUATION

Definition. An equation f(x, y, \(\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}, \ldots \frac{d^{n} y}{d x^{n}}\)) = 0, which expresses a relation between dependent and independent variables and their derivatives of any order, is called a differential equation.

2. FORMATION OF DIFFERENTIAL EQUATIONS

Here we differentiate the given equation as many times as the number of arbitrary constants and then eliminate the arbitrary constants from them.

3. SOLUTION OF DIFFERENTIAL EQUATION

It is a relation between the variables involved such that this relation and the differential co-efficients obtained therefrom satisfy the given differential equation.

4. SOLUTION OF DIFFERENT FORMS OF DIFFERENTIAL EQUATIONS

(i) If the equation is :

\(\frac{d y}{d x}\) = f(x), then y = ∫ f(x)dx + c.

(ii) Variables Separable. If the equation is:

\(\frac{d y}{d x}\) = f(x)g(y), then: \(\int \frac{d y}{g(y)}\) = ∫ f(x) + c.

(iii) Reducible to Variables Seperable.

If the equation is \(\frac{d y}{d x}\) = f(ax + by + c), then put ax + by + c= z. dx

(iv) Homogeneous Equation.

If the equation is \(\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)}\), where f(x, y), g(x, y) are homogeneous functions of the degree in x and v. then put y = vx.

(v) Linear Equation.

If the equation is \(\frac{d y}{d x}\) + Px = Q where P, Q are constants or functions of x, then ye^{∫Pdx} = ∫Qe^{∫Pdx} dx + c, where e^{∫Pdx} is the integrating factor (I.F.).