Differential Equations – Chapter 9

A fun, mobile-friendly practice worksheet for students. Build confidence in order & degree, formation, variable separation, homogeneous equations, linear differential equations and applications.

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📘 Differential Equations Class 12 – Complete Worksheet Practice Guide

Differential equations are an important part of Class 12 Mathematics because they connect a quantity with its rate of change. This online worksheet is designed for repeated practice of the main ideas used in the chapter. The questions are generated with changing numerical values so students can start a fresh test instead of memorising one fixed worksheet.

1. What is a Differential Equation?

A differential equation is an equation involving a dependent variable, an independent variable and derivatives of the dependent variable. For example, dy/dx = 3x² relates y and its derivative. A differential equation can describe growth, motion, curves and many real-life relationships.

2. Order and Degree

The order is the order of the highest derivative present in the differential equation. The degree is the power of the highest order derivative when the equation is polynomial in derivatives.

⭐ Formula idea: Order = highest order derivative present.

⭐ Degree = power of the highest order derivative (after removing radicals/fractions involving derivatives where applicable).

3. Formation of Differential Equations

A differential equation can be formed by eliminating arbitrary constants from a family of curves. If a relation contains one arbitrary constant, differentiating once is commonly enough to eliminate it. With two arbitrary constants, two independent differentiations or suitable elimination steps may be required.

4. Variable Separable Differential Equations

When a differential equation can be rearranged so that terms containing y occur with dy and terms containing x occur with dx, it is called a variable separable form.

⭐ If dy/dx = f(x)g(y), then dy/g(y) = f(x) dx.

Integrate both sides and remember to include the constant of integration. Initial conditions can then be used to find the particular solution.

5. Homogeneous Differential Equations

A first-order differential equation of the form dy/dx = F(y/x) can often be handled by putting y = vx. Then dy/dx = v + x dv/dx. The substitution reduces the equation to a variable separable equation in v and x.

⭐ Put y = vx; therefore dy/dx = v + x(dv/dx).

6. Linear Differential Equations

A first-order linear differential equation is commonly written as dy/dx + P(x)y = Q(x). Its solution uses the integrating factor.

⭐ Integrating Factor (IF) = e^∫P(x)dx.

⭐ Solution: y × IF = ∫ Q(x) × IF dx + C.

For questions in the worksheet, students should identify P(x) and Q(x), calculate the integrating factor, multiply throughout by the IF, and integrate.

7. General and Particular Solutions

A general solution contains an arbitrary constant, usually written as C. When an initial or boundary condition is supplied, substitution determines the constant and gives a particular solution. Always check whether the obtained solution satisfies the original equation and condition.

8. Applications and Interpretation

Differential equations are useful when a rate of change is known. For example, if the rate is proportional to the quantity itself, separation of variables can lead to exponential-type solutions. In school-level applications, focus on translating the statement into a differential equation before solving it.

9. How to Use This DigiSadhan Worksheet

Choose the number of questions, question mode, topic, difficulty and timer. Press Start New Test. Each question provides a checking button. For short-answer questions, type the answer in the box and press Check Answer. The answer remains hidden until you check the question. Correct answers turn green and incorrect answers turn red. The progress bar and score update as you work.

Use Shuffle Whole Worksheet to create a new order. For a fresh set, press Start New Test. The worksheet supports up to 2,000 questions in one test. Use sound controls if you want feedback sounds, and select a timer when starting a new test for exam-style practice.

10. Smart Revision Tips

Memorise the standard linear form and integrating factor.
Practise identifying order and degree quickly.
For separable equations, separate variables before integrating.
For homogeneous equations, test whether the right side can be expressed in y/x and try y = vx.
Keep the constant of integration in general solutions.
Use initial conditions only after obtaining the general solution unless the question structure suggests otherwise.
Check signs carefully when integrating.
Use the mixed mode regularly to simulate exam practice.

11. Formula Quick Revision

⭐ dy/dx + P(x)y = Q(x)

⭐ IF = e^∫P(x)dx

⭐ y(IF) = ∫Q(x)(IF)dx + C

⭐ Homogeneous substitution: y = vx

⭐ dy/dx = v + x dv/dx

⭐ Separable pattern: dy/g(y) = f(x)dx

12. Exam Practice Strategy

Begin with easy identification questions to warm up. Then practise standard solution methods. Once comfortable, select Hard difficulty and Mixed mode. After completing a test, review every incorrect response rather than only looking at the final score. Repeating shuffled tests helps build recognition of methods and reduces dependence on memorised question order.

Important: This is an independent DigiSadhan practice resource inspired by the chapter topic. The supplied NCERT URL is the official NCERT textbook portal, which identifies itself as “Textbooks PDF (I-XII)”. citeturn0view0