Differential Equations Summary Notes

In this article, we will study some basic concepts related to differential equations. Further, we will discuss some methods to find their general and particular solutions.

Let’s first know the meaning of such an equation.

1. Basics – What is a Differential Equation?

It is an equation that involves the derivative(s) of dependent variable with respect to the independent variable(s). It tells us how a quantity changes rather than just its value.

For example,
$\frac{dy}{dx} + y = x$ Here, besides having x and y, we also have the derivative dy/dx.

Higher-order derivatives like
$\frac{d^2y}{dx^2}, \quad \frac{d^3y}{dx^3}$ can also appear.

This is unlike a normal equation or relation involving just the variables x and/or y, for instance, x + y = 7, which is not a differential equation.

2. Order and Degree of a Differential Equation

Order: It is the highest derivative of the dependent variable with respect to the independent variable present in the equation.

Degree: The exponent of the highest order derivative (only if it’s a polynomial in derivatives).

Degree is not defined, if it is not a polynomial in derivatives as you can see the examples in the image below:

Order and Degree of Differential Eqn. — Watch it on YouTube

3. Formation of Differential Equations

We can form a differential equation from a given family of curves by differentiating it as many times as the number of arbitrary constants in it. And, the idea is to eliminate all the constants to get the required differential equation.

Example: The equation of a family of circles centered at the origin is:
$x^2 + y^2 = r^2$
Differentiating both sides, we get,
$2x + 2yy' = 0$ ,
which is the required differential equation representing all such circles.

⬇️ Download the handwritten PDF for better understanding with an example.

4. Solution of Differential Equations

General Solution: It contains arbitrary constants.

Particular Solution: It is found by substituting given conditions into the general solution and thus eliminating the constants.

Remember: The number of constants in a general solution is equal to the order of the differential equation. Whereas, a particular solution does not have a constant.

For example:
To solve $\frac{dy}{dx} = 2x$ , we integrate and get,
$y = x^2 + C$
Here, C is an arbitrary constant. If we have a condition like y = 5 when x = 1, i.e. y(1) = 5, we substitute and solve for C, giving a particular solution.

Like in this case, after putting the given values of x and y, we get 5 = 1 + C means C = 4.

So the particular solution will be y = x² + 4, which represents a parabola having vertex at (0, 4) and opens upwards.

5. Variable Separable Method

If we can write the equation as a product of two functions, one in x and one in y, we separate them and integrate.

Steps:

Express in the form f(y) dy = g(x) dx
Integrate both sides to solve.

Example:
$\frac{dy}{dx} = \frac{x}{y}$

Rewriting the equation, we get:

$y dy = x dx$

Integrating both sides, we get:

$y^2 = x^2 + C$

This is the required general solution.

⬇️ You can download the PDF for quick notes on the different types of differential equations.

6. What is a Homogeneous Function?

A function f(x, y) is a homogeneous function of degree n if:

$f(\lambda x, \lambda y) = \lambda^n f(x, y)$

Example:
$f(x, y) = x^3 + y^3$

Checking homogeneity:
$f(\lambda x, \lambda y) = (\lambda x)^3 + (\lambda y)^3 = \lambda^3 (x^3 + y^3) = \lambda^3 f(x, y)$

Since the function satisfies the above condition for n = 3, it is homogeneous function of degree 3.

⬇️ Checkout for more examples in the attached handwritten PDF here.

7. Homogeneous Differential Equations

A homogeneous differential equation is one which can be expressed in the follwing forms:

$\frac{dy}{dx} = f(y/x)$ or

$\frac{dx}{dy} = f(x/y)$

For example: $\frac{dy}{dx} = \frac{x + y}{x}$

These equations can be solved by substituting y/x = v or x/y = v resp., depending upon the type as explained in the image below:

Homogeneous Differential Equations — Watch it on YouTube

8. Linear Differential Equations

A first-order linear differential equation is of the form:

$\frac{dy}{dx} + P y = Q$

where P and Q are functions of x or constants.

Solution involves the Integrating Factor (I.F.):

$I.F. = e^{\int P dx}$

Similarly, for linear differential equations of the type,

$\frac{dx}{dy} + P x = Q$ ,

where P and Q are functions of y or constants.

➡️ Check out the handwritten PDF for step-by-step solutions!

9. Word Problems on Differential Equations

These equations arise in different applications, be it Physics, Chemistry, Biology, Anthropology, Geology, Economics etc.

Many real-world problems, like population growth, radioactive decay, and cooling, are modeled using differential equations. Hence, an indepth study of these equations is very important in all modern scientific investigations.

10. Miscellaneous Concepts

Ordinary Differential Equations (ODEs): Only one independent variable.

Partial Differential Equations (PDEs): More than one independent variable. (Not in syllabus (Class 12, CBSE 2025) but good to know!)

Linear vs. Non-Linear Equations: If derivatives appear only in first power, it’s linear; otherwise, it’s non-linear. There are other ways to identify a non-linear differential equation as mentioned in the PDF.

Final Thoughts

Differential equations are everywhere in Math, Physics and Engineering. The key is to understand how they describe change and master the different methods to solve them.

✅ For a better understanding and quick notes for the revision during exams, download the handwritten PDF with step-by-step explanations, just like in the previous topic on Linear Programming for Class 12.

DIFFERENTIAL EQUATIONS CONCEPTS (pdf)Download