JEE (Main) Mathematics - Differential Equations MCQs

16. The differential equation:

\(\frac{dy}{dx} = y \cdot \sin(x)\)

1. \(y = C \cdot e^{\sin(x)}\)
2. \(y = C \cdot e^{-\cos(x)}\)
3. \(y = C \cdot e^{\cos(x)}\)
4. \(y = C \cdot \sin(x)\)

Answer: B

17. Solve the differential equation:

\(\frac{dy}{dx} = x \cdot y\)

1. \(y = Ce^x\)
2. \(y = Ce^{x^2/2}\)
3. \(y = C \cdot x^2\)
4. \(y = C \cdot \ln(x)\)

Answer: B

18. The solution of the differential equation:

\(\frac{dy}{dx} = \frac{x}{y}\)

1. \(y^2 = x^2 + C\)
2. \(y = \sqrt{x + C}\)
3. \(y^2 = x + C\)
4. \(y = x + C\)

Answer: A

19. The differential equation:

\(\frac{dy}{dx} = e^x \cdot e^y\)

1. \(\frac{dy}{e^y} = e^x \, dx\)
2. \(e^{-y} \, dy = e^x \, dx\)
3. \(e^y \, dy = e^x \, dx\)
4. Both A and B

Answer: D

20. Solve the equation:

\(\frac{dy}{dx} = \frac{1}{y^2}\)

1. \(y = \sqrt{x + C}\)
2. \(y = \sqrt[3]{x + C}\)
3. \(y = \frac{1}{\sqrt{x + C}}\)
4. \(y = x + C\)

Answer: C

21. The general solution of the equation:

\(\frac{dy}{dx} = x^2 \cdot y^3\)

1. \(y^2 = \frac{1}{x^3 + C}\)
2. \(y^2 = \frac{1}{x^3 - C}\)
3. \(y^2 = \frac{1}{2x^3 + C}\)
4. \(y^2 = \frac{1}{2x^3 - C}\)

Answer: C

22. Which of the following is separable?

1. \(\frac{dy}{dx} + y = x\)
2. \(\frac{dy}{dx} = x \cdot y\)
3. \(\frac{dy}{dx} = x + y\)
4. \(y = \sin(x) \cdot \frac{dy}{dx}\)

Answer: B

23. Solve \(\frac{dy}{dx} = \frac{\ln(x)}{y}\).

1. \(y^2 = 2\ln(x) + C\)
2. \(y^2 = \ln^2(x) + C\)
3. \(y = 2\ln(x) + C\)
4. \(y = \ln^2(x) + C\)

Answer: A

24. Find the solution of \(\frac{dy}{dx} = x^2 + y^2\).

1. \(y = \tan^{-1}(x) + C\)
2. \(y = \cot^{-1}(x) + C\)
3. \(y^2 + x^2 = C\)
4. \(y^2 - x^2 = C\)

Answer: C

25. Solve \(\frac{dy}{dx} = \frac{1 - y^2}{1 + x^2}\).

1. \(y = \sin^{-1}(x)\)
2. \(y = \cos^{-1}(x)\)
3. \(y = \tan^{-1}(x)\)
4. \(y = \cot^{-1}(x)\)

Answer: A

26. Solve the differential equation \(\frac{dy}{dx} = y(1 - y)\):

1. \(y = \frac{1}{1 + e^{-x}}\)
2. \(y = \frac{1}{1 - e^{-x}}\)
3. \(y = e^x\)
4. \(y = \ln(x)\)

Answer: A