Class 12 Maths Chapter 9 Differential Equations – Exercise 9.2 NCERT Solutions

Introduction

Exercise 9.2 focuses on general and particular solutions of differential equations. Students learn how to integrate given differential equations, apply initial conditions, and distinguish between general and particular solutions. This exercise is crucial for solving real‑life problems modeled by differential equations.

Key Concepts

1. General Solution: Contains arbitrary constant(s). Example: $\frac{dy}{dx}=2x\Rightarrow y=x^2+C$.

2. Particular Solution: Obtained by applying initial condition(s). Example: If $y(0)=1$, then $y=x^2+1$.

3. Method:

   • Separate variables if possible.

   • Integrate both sides.

   • Apply initial condition to find constant(s).

Students Frequently Make Mistakes

• Forgetting to add constant of integration in general solution.

• Misapplying initial condition.

• Confusing general and particular solutions.

• Errors in integration of trigonometric/exponential functions.

• Skipping domain restrictions.

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

Q1. Solve $\frac{dy}{dx}=2x$.

$$y=\int 2xdx=x^2+C$$

Q2. Solve $\frac{dy}{dx}=3x^2$.

$$y=\int 3x^{2}dx=x^3+C$$

Q3. Solve $\frac{dy}{dx}=e^x$.

$$y=\int e^{x}dx=e^x+C$$

Q4. Solve $\frac{dy}{dx}=\cos x$.

$$y=\int \cos xdx=\sin x+C$$

Q5. Solve $\frac{dy}{dx}=\sin x$.

$$y=\int \sin xdx=-\cos x+C$$

Q6. Solve $\frac{dy}{dx}=\frac{1}{x}$.

$$y=\int \frac{1}{x}dx=\ln \mathrm{\mid }x\mathrm{\mid }+C$$

Q7. Solve $\frac{dy}{dx}=x^2+1$.

$$y=\int (x^2+1)dx=\frac{x^3}{3}+x+C$$

Q8. Solve $\frac{dy}{dx}=2x+3$.

$$y=\int (2x+3)dx=x^2+3x+C$$

Q9. Solve $\frac{dy}{dx}={\sec }^{2}x$.

$$y=\int {\sec }^{2}xdx=\tan x+C$$

Q10. Solve $\frac{dy}{dx}={\csc }^{2}x$.

$$y=\int {\csc }^{2}xdx=-\cot x+C$$

FAQs (10)

FAQ1. What is general solution? Solution with arbitrary constant(s).

FAQ2. What is particular solution? Solution after applying initial condition(s).

FAQ3. Why add constant of integration? Because indefinite integral represents family of functions.

FAQ4. What is solution of $\frac{dy}{dx}=2x$? $y=x^2+C$.

FAQ5. What is solution of $\frac{dy}{dx}=3x^2$? $y=x^3+C$.

FAQ6. What is solution of $\frac{dy}{dx}=e^x$? $y=e^x+C$.

FAQ7. What is solution of $\frac{dy}{dx}=\cos x$? $y=\sin x+C$.

FAQ8. What is solution of $\frac{dy}{dx}=\sin x$? $y=-\cos x+C$.

FAQ9. What is solution of $\frac{dy}{dx}=\frac{1}{x}$? $y=\ln \mathrm{\mid }x\mathrm{\mid }+C$.

FAQ10. Why is Exercise 9.2 important? It builds foundation for solving differential equations with general and particular solutions.

Conclusion

Exercise 9.2 has 10 solved questions and 10 FAQs that strengthen your understanding of general and particular solutions of differential equations. This builds the foundation for advanced problem solving in Class 12 Maths.

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