Maharashtra Board Class 12 Maths Commerce Part I Chapter 8 Differential Equation and Applications PDF Download

Differential Equations and Applications

Let's Study

Differential Equation

Ordinary differential equation

Order and degree of a differential equation

Solution of a differential equation

Formation of a differential equation

Applications of differential equations

Let's Recall

Independent variable

Dependent variable

Equation

Derivatives

Integration

Let's Learn

8.1 Differential Equations

Definition

An equation involving dependent variable(s), independent variable and derivative(s) of dependent variable(s) with respect to the independent variable is called a differential equation.

For example:

\(\frac{dy}{dx} + y = x\)

\(x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0\)

\(\frac{d^2y}{dt^2} = 2t\)

\(r \frac{dr}{d\theta} + e^\theta = 8\)

\(\sqrt{1 + \frac{dy}{dx}} = \frac{d^2y}{dx^2}\)

\(x \, dx + y \, dy = 0\)

8.1.1 Ordinary Differential Equation

A differential equation in which the dependent variable, say y, depends only on one independent variable, say x, is called an ordinary differential equation.

8.1.2 Order Of A Differential Equation

It is the order of the highest order derivative occurring in the differential equation.

\(\frac{dy}{dx} + y = x\) is of order 1

\(x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0\) is of order 2

\(\left(\frac{d^2y}{dx^2}\right)^2 + x \frac{dy}{dx} = 2y\) is of order 2

\(\frac{2dy}{dx} = e^x\) is of order 1

8.1.3 Degree Of A Differential Equation

It is the power of the highest order derivative when all the derivatives are made free from fractional indices and negative sign, if any.

For example:

\(x^2 \left(\frac{d^2y}{dx^2}\right) + x \frac{dy}{dx} + y = 0\)

In this equation, the highest order derivative is \(\frac{d^2y}{dx^2}\) and its power is one. Therefore this equation has degree one.

Teacher's Note

Order tells us which derivative is highest. Degree tells us the power of that highest derivative. Like in a school, order is the class number and degree is how many times the student appears.

Exam Trick

Remember: First remove all roots and negative signs. Then find the power of the highest derivative. That power is the degree. Order is always just the number of the derivative.

Points to Remember

Order is the highest derivative in the equation.
Degree is the power of that highest derivative.
Always remove roots and negative signs before finding degree.
Order and degree are always positive whole numbers.
Order is never the same as degree if there is a root.

Solution Of A Differential Equation

A function of the form \(y = f(x) + c\) which satisfies the given differential equation is called the solution of the differential equation.

Every differential equation has two types of solutions: 1) General and 2) Particular

1) General Solution

A solution of the differential equation in which the number of arbitrary constants is equal to order of differential equation is called a general solution.

2) Particular Solution

A solution of the differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution.

Solved Examples

1) Verify that the function \(y = ae^x + be^{2x}\), where \(a, b \in \mathbb{R}\) is a solution of the differential equation \(\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0\).

Solution: Consider the function

\(y = ae^x + be^{2x}\) .......... (I)

Differentiating both sides of equation I with respect to x, we get

\(\frac{dy}{dx} = ae^x + 2be^{2x}\) .......... II and

Differentiating both sides of equation II with respect to x, we get

\(\frac{d^2y}{dx^2} = ae^x + 4be^{2x}\) .......... III

Now, L.H.S = \(\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y\)

= \(ae^x + 4be^{2x} - (ae^x + 2be^{2x}) - 2(ae^x + be^{2x})\) from II and III

= \(2ae^x + 2be^{2x}\)

= \(2(ae^x + be^{2x}) - 2y\) from I

= R.H.S

Therefore, the given function is a general solution of the given differential equation.

Teacher's Note

To verify a solution, differentiate it as many times as needed, then put all values in the equation. If both sides match, it is correct. Like checking if a student's homework answer is right.

Exam Trick

Count the arbitrary constants in the solution. If there are n constants and the equation is order n, then it is a general solution. If constants have values, it is a particular solution.

Points to Remember

General solution has arbitrary constants.
Particular solution has no arbitrary constants.
Number of constants equals the order of the equation for general solution.
To verify, differentiate and substitute back in the equation.
Both sides of equation must be equal after verification.

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