CBSE Class 12 Mathematics Differential Equations Assignment Set 08

Practice Questions with Answers

Question 1. Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0 \).
Answer:
Order 2 and degree 4

Question 2. Find the integrating factor of the differential equation \( \cos x\frac{dy}{dx} + y\sin x = 1 \).
Answer:
sec x

Question 3. Find the particular solution of \( \frac{dy}{dx} = \frac{y + 1}{x + 1} \), when y(1) = 2.
Answer:
\( y = \frac{3}{2}x + \frac{1}{2} \)

Question 4. Find the solution of the differential equation cos x sin y dx + sin x cos y dy = 0.
Answer:
sin x sin y = C

Question 5. Find the particular solution of \( \frac{dy}{dx} + y = e^{-x} \), y(0) = 0.
Answer:
\( y = xe^{-x} \)

Question 6. Find the integrating factor of the differential equation \( (1 - x^2)\frac{dy}{dx} - xy = 1 \).
Answer:
\( \sqrt{1 - x^2} \)

Question 7. Find the solution of \( x\frac{dy}{dx} + y = e^x \).
Answer:
\( y = \frac{e^x}{x} + \frac{C}{x} \)

Question 8. Find the solution of the differential equation \( x\frac{dy}{dx} + 2y = x^2 \).
Answer:
\( y = \frac{x^2}{4} + Cx^{-2} \)

Question 9. Find the integrating factor of the differential equation \( \frac{dy}{dx} + y = \frac{1 + y}{x} \).
Answer:
\( \frac{e^x}{x} \)

Question 10. Find the degree of the differential equation \( \frac{d^2y}{dx^2} + e^{dy/dx} = 0 \).
Answer:
Not defined

Question 11. Verify that \( x + y = \tan^{-1} y \) is a solution of the differential equation \( y^2y' + y^2 + 1 = 0 \).
Answer:
Differentiating, \( 1 + y' = \frac{y'}{1 + y^2} \) ⇒ \( (1 + y^2) + y'(1 + y^2) = y' \) ⇒ \( y^2y' + y^2 + 1 = 0 \). Hence verified.

Question 12. Find the solution of the differential equation \( \frac{dy}{dx} = e^{-2x} \).
Answer:
\( 2y + e^{-2x} = C \)

Question 13. Find the solution of the following differential equation: \( x\sqrt{1 + y^2}\,dx + y\sqrt{1 + x^2}\,dy = 0 \)
Answer:
\( \sqrt{1 + x^2} + \sqrt{1 + y^2} = C \)

Question 14. Find the sum of the order and degree of the differential equation \( \left(1 + 3\frac{dy}{dx}\right)^{2/3} = 4\frac{d^3y}{dx^3} \).
Answer:
6

Question 15. Find the general solution of the differential equation \( \frac{dy}{dx} = \cot x\cot y \).
Answer:
sin x cos y = C