Chapter 9, Differential Equations, is one of the most important chapters of calculus in Class 12. This chapter explains how to form and solve equations involving derivatives. It connects differentiation and integration and shows how mathematics is used to study change in real-life situations.
This chapter has fixed question patterns, so with proper practice, students can score very well. At Edu Tehri, this chapter is explained with step-by-step methods, handwritten notes, solved NCERT exercises, and free PDFs.
• A function
• One or more of its derivatives
In simple words, it shows the relationship between a variable and its rate of change.
• Growth of population
• Motion of objects
• Cooling and heating processes
• Spread of diseases
• Electrical circuits
This chapter helps students understand how mathematics explains continuous change.
The order of a differential equation is defined as the highest order derivative present in the equation.
In Class 12, NCERT focuses mainly on first-order differential equations.
The degree of a differential equation is the power of the highest order derivative, provided the equation is polynomial in derivatives.
A solution of a differential equation is a function that satisfies the given equation.
This function is a solution because its derivative gives the original equation.
A general solution of a differential equation contains an arbitrary constant (C). It represents a family of solutions.
A particular solution is obtained by assigning a specific value to the constant using a given condition.
Differential equations can be formed by eliminating constants from given equations.
1. Write the given equation
2. Differentiate it
3. Eliminate constants
This topic is theory-based and usually asked as short answer questions.
1. Variable separable
2. Homogeneous
3. Linear differential equations
A differential equation is called variable separable if variables x and y can be separated on opposite sides.
1. Separate variables
2. Integrate both sides
3. Add constant of integration
\ frac {dy} {dx} = F \ left (\frac{y} {x}\right)
1. Put y = vx
2. Then \frac{dy}{dx} = v + x\frac {dv}{dx}
3. Substitute and simplify
4. Convert into separable form
5. Integrate
Homogeneous equations are frequently asked in board exams.
\frac{dy}{dx} + Py = Q
Where P and Q are functions of x.
1. Identify P and Q
2. Find Integrating Factor (IF):
IF = e^{\int P dx}
3. Multiply equation by IF
4. Integrate both sides
5. Find solution
This method is very important for 5-mark questions.
1. General solution: y = f(x) + C
2. Homogeneous substitution: y = vx
3. Linear equation form: \frac {dy}{dx} + Py = Q
4. Integrating factor: IF = e^{\int P dx}
NCERT Chapter 9 contains 5 exercises.
Download Chapter-9 with Exercises - Free PDF • Order and degree
• Formation of equations
• Variable separable equations
• Homogeneous equations
• Linear differential equations
• Usually asked as 4–5 mark questions
• Patterns are repeated every year
• Very scoring chapter if methods are clear
• Free NCERT solutions PDF
• Handwritten step-by-step notes
• Formula sheets
• Board-focused practice questions
• Easy explanation videos (if applicable)
• Connects differentiation and integration
• Real-life applications
• Fixed exam pattern
• Concept + practice based
• Highly scoring
A differential equation is an equation that involves a function and its derivative. In simple words, it shows the relationship between a variable and its rate of change.
dy\dx = x
is a differential equation because it contains the derivative of y with respect to x.
Differential equations are used to describe real-life situations such as growth and decay, motion, population changes, and temperature variation.
The order of a differential equation is the highest order of derivative present in the equation.
• dy \ dx = x → Order = 1
• d2y \ dx2 + y = 0 → Order = 2
In Class 12 NCERT, students mainly study first-order differential equations.
The degree of a differential equation is the power of the highest order derivative, after removing fractions and radicals involving derivatives.
(dy \ dx)2 + y = 0
Here, degree = 2.
Degree is defined only when the equation is polynomial in derivatives.
A solution of a differential equation is a function that satisfies the given differential equation.
if
dy\dx = 2x
then
y = x^2 + C
is a solution, because its derivative gives back 2x.
In exams, students are mainly asked to find the general or particular solution.
A general solution of a differential equation contains an arbitrary constant (C).
It represents a family of solutions, not just one.
\frac{dy}{dx} = x
General solution:
y = \frac {x^2} {2} + C
General solutions are important because they show all possible solutions of the differential equation.
A particular solution is obtained from the general solution by using a given condition (also called initial condition).
If
y = \frac {x^2} {2} + C
and given that y = 1 when x = 0, then
C = 1
So the particular solution becomes:
y = \frac{x^2}{2} + 1
A separable differential equation is one in which variables x and y can be separated on opposite sides of the equation.
\ frac {dy} {dx} = f(x) g(y)
1. Separate x and y terms
2. Integrate both sides
3. Add constant of integration
This is the most important and most asked type in board exams.
\frac {dy} {dx} = F\left (\frac {y} {x} \right)
• Put y = vx
• Convert equation into separable form
• Integrate
Homogeneous differential equations are frequently asked in 4–5 mark questions.