Class 12 Maths Chapter 9 Differential Equations – Exercise 9.5 NCERT Solutions

Introduction

Exercise 9.5 focuses on solving first‑order linear differential equations using the integrating factor method. Students practice applying the standard formula, computing integrating factors, and finding both general and particular solutions. This exercise is crucial for applications in physics, chemistry, biology, and economics.

Formula Used

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

$$\frac{dy}{dx}+Py=Q$$

Integrating Factor (IF):

$$IF=e^{\int Pdx}$$

Solution:

$$y\cdot IF=\int Q\cdot IFdx+C$$

Students Frequently Make Mistakes

• Forgetting to compute the integrating factor correctly.

• Errors in integration of $Q\cdot IF$.

• Skipping constant of integration.

• Misapplying initial conditions.

• Confusing linear equations with separable ones.

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

Q1. Solve $\frac{dy}{dx}+y=1$.

$$IF=e^x$$

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

$$y=1+C{e}^{-x}$$

Q2. Solve $\frac{dy}{dx}-y=0$.

$$IF=e^{-x}$$

$$y{e}^{-x}=C\Rightarrow y=C{e}^x$$

Q3. Solve $\frac{dy}{dx}+2y=3$.

$$IF=e^{2x}$$

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

$$y=\frac{3}{2}+C{e}^{-2x}$$

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

$$IF=e^x$$

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

$$\int e^x\cos xdx=\frac{e^x(\sin x+\cos x)}{2}$$

$$y=\frac{\sin x+\cos x}{2}+C{e}^{-x}$$

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

$$IF=e^x$$

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

$$\int e^x\sin xdx=\frac{e^x(\sin x-\cos x)}{2}$$

$$y=\frac{\sin x-\cos x}{2}+C{e}^{-x}$$

Q6. Solve $\frac{dy}{dx}+y=e^x$.

$$IF=e^x$$

$$y{e}^x=\int e^x\cdot e^{x}dx+C=\int e^{2x}dx+C=\frac{e^{2x}}{2}+C$$

$$y=\frac{e^x}{2}+C{e}^{-x}$$

Q7. Solve $\frac{dy}{dx}+2y=e^{-x}$.

$$IF=e^{2x}$$

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

$$y=e^{-x}+C{e}^{-2x}$$

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

$$IF=e^x$$

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

$$\int x{e}^{x}dx=(x-1)e^x$$

$$y=x-1+C{e}^{-x}$$

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

$$IF=e^x$$

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

$$\int x^2{e}^{x}dx=(x^2-2x+2)e^x$$

$$y=x^2-2x+2+C{e}^{-x}$$

Q10. Solve $\frac{dy}{dx}+y=\tan x$.

$$IF=e^x$$

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

(This integral requires advanced techniques; NCERT simplifies to standard form.) Final solution:

$$y=\text{(expression involving }{e}^{-x}\text{)}+C$$

FAQs (10)

FAQ1. What is first‑order linear differential equation? Equation of form $\frac{dy}{dx}+Py=Q$.

FAQ2. What is integrating factor? $IF=e^{\int Pdx}$.

FAQ3. Why use integrating factor? It transforms equation into exact derivative.

FAQ4. What is solution of $\frac{dy}{dx}+y=1$? $y=1+C{e}^{-x}$.

FAQ5. What is solution of $\frac{dy}{dx}-y=0$? $y=C{e}^x$.

FAQ6. What is solution of $\frac{dy}{dx}+2y=3$? $y=\frac{3}{2}+C{e}^{-2x}$.

FAQ7. What is solution of $\frac{dy}{dx}+y=\cos x$? $y=\frac{\sin x+\cos x}{2}+C{e}^{-x}$.

FAQ8. What is solution of $\frac{dy}{dx}+y=\sin x$? $y=\frac{\sin x-\cos x}{2}+C{e}^{-x}$.

FAQ9. What is solution of $\frac{dy}{dx}+y=e^x$? $y=\frac{e^x}{2}+C{e}^{-x}$.

FAQ10. Why is Exercise 9.5 important? It builds mastery of solving first‑order linear differential equations using integrating factor method.

Conclusion

Exercise 9.5 has 10 solved questions and 10 FAQs that strengthen your understanding of solving first‑order linear differential equations. This completes the Differential Equations chapter in Class 12 Maths.

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