Class 12 Maths Chapter 9 Differential Equations

Class 12 Maths Chapter 9 Differential Equations – Exercise 9.1 NCERT Solutions

Introduction

Exercise 9.1 introduces the basic concepts of differential equations. Students learn definitions, order, degree, and how to form differential equations from given relations. This exercise builds the foundation for solving and applying differential equations in later sections.

Key Concepts

Differential Equation: An equation involving derivatives of a function.
Order of Differential Equation: The highest order derivative present.
Degree of Differential Equation: The power of the highest order derivative (when equation is polynomial in derivatives).
Formation of Differential Equation: Eliminate arbitrary constants from a relation using differentiation.

Students Frequently Make Mistakes

Confusing order with degree.
Forgetting to check if equation is polynomial in derivatives before defining degree.
Errors in eliminating constants.
Misinterpreting derivative notation.
Skipping steps in formation process.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find order and degree of $\frac{d^{2} y}{d x^{2}} + y = 0$ . Order = 2, Degree = 1.

Q2. Find order and degree of ${(\frac{d y}{d x})}^{2} + y = 0$ . Order = 1, Degree = 2.

Q3. Find order and degree of $\frac{d^{3} y}{d x^{3}} + x \frac{d y}{d x} = 0$ . Order = 3, Degree = 1.

Q4. Find order and degree of $\sin (\frac{d y}{d x}) + y = 0$ . Not polynomial in derivative ⇒ Degree not defined. Order = 1.

Q5. Form differential equation from $y = c x + d$ . Differentiate: $\frac{d y}{d x} = c$ . Eliminate constants ⇒ $\frac{d^{2} y}{d x^{2}} = 0$ .

Q6. Form differential equation from $y = m x + c$ . Differentiate: $\frac{d y}{d x} = m$ . Eliminate constants ⇒ $\frac{d^{2} y}{d x^{2}} = 0$ .

Q7. Form differential equation from $y = A e^{x} + B e^{- x}$ . Differentiate: $\frac{d y}{d x} = A e^{x} - B e^{- x}$ . $\frac{d^{2} y}{d x^{2}} = A e^{x} + B e^{- x} = y$ . Equation: $\frac{d^{2} y}{d x^{2}} = y$ .

Q8. Form differential equation from $y = A \cos x + B \sin x$ . Differentiate: $\frac{d y}{d x} = - A \sin x + B \cos x$ . $\frac{d^{2} y}{d x^{2}} = - A \cos x - B \sin x = - y$ . Equation: $\frac{d^{2} y}{d x^{2}} + y = 0$ .

Q9. Form differential equation from $y = A e^{2 x} + B e^{- 2 x}$ . Differentiate: $\frac{d y}{d x} = 2 A e^{2 x} - 2 B e^{- 2 x}$ . $\frac{d^{2} y}{d x^{2}} = 4 A e^{2 x} + 4 B e^{- 2 x} = 4 y$ . Equation: $\frac{d^{2} y}{d x^{2}} = 4 y$ .

Q10. Form differential equation from $y = A \cos 2 x + B \sin 2 x$ . Differentiate: $\frac{d y}{d x} = - 2 A \sin 2 x + 2 B \cos 2 x$ . $\frac{d^{2} y}{d x^{2}} = - 4 A \cos 2 x - 4 B \sin 2 x = - 4 y$ . Equation: $\frac{d^{2} y}{d x^{2}} + 4 y = 0$ .

FAQs (10)

FAQ1. What is differential equation? Equation involving derivatives of a function.

FAQ2. What is order of differential equation? Highest order derivative present.

FAQ3. What is degree of differential equation? Power of highest order derivative (if polynomial).

FAQ4. When is degree not defined? If equation is not polynomial in derivatives.

FAQ5. How to form differential equation? Eliminate constants using differentiation.

FAQ6. What is order of $\frac{d^{2} y}{d x^{2}} + y = 0$ ?

FAQ7. What is degree of ${(\frac{d y}{d x})}^{2} + y = 0$ ?

FAQ8. What is equation formed from $y = A e^{x} + B e^{- x}$ ? $\frac{d^{2} y}{d x^{2}} = y$ .

FAQ9. What is equation formed from $y = A \cos x + B \sin x$ ? $\frac{d^{2} y}{d x^{2}} + y = 0$ .

FAQ10. Why is Exercise 9.1 important? It builds foundation for solving differential equations.

Conclusion

Exercise 9.1 has 10 solved questions and 10 FAQs that strengthen your understanding of order, degree, and formation of differential equations. This begins the Differential Equations chapter in Class 12 Maths.

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