linear differential equation (introduction & solution ) ,linear dependence and its solution MCQs

1. What is the order of the differential equation $\frac{d^{2}}{y} + 3 \frac{dy}{dx} + 2 y = 0$ ?

a) 0 b) 1 c) 2 d) 3

Correct Answer: c) 2

Explanation: The order of a differential equation is the highest derivative present. Here, the highest derivative is d²y/dx², making it 2nd order.

2. Which of the following is a linear differential equation?

y^{''} + y^{2} = 0

y y^{'} + x = 0

y^{'} + 2 y = e^{x}

y^{'} + \sin y = 0

Correct Answer: c) $y^{'} + 2 y = e^{x}$

Explanation: A linear differential equation has dependent variable (y) and its derivatives in first degree only. Option c satisfies this condition.

3. What is the general solution of $\frac{dy}{dx} + 2 y = 0$ ?

y = C e^{2 x}

y = C e^{- 2 x}

y = 2 C e^{x}

y = C e^{- x}

Correct Answer: b) $y = C e^{- 2 x}$

Explanation: This is a first-order linear ODE with solution $y = C e^{- P (x) x}$ where P(x)=2.

4. The integrating factor for $\frac{dy}{dx} + 3 x^{2} y = x^{2}$ is:

e^{x^{3}}

e^{3 x^{2}}

e^{3 x}

e^{x^{2}}

Correct Answer: a) $e^{x^{3}}$

Explanation: The integrating factor is $e^{\int 3 x^{2} dx} = e^{x^{3}}$ .

5. The functions $f (x) = e^{x}$ and $g (x) = 2 e^{x}$ are:

a) Linearly independent b) Linearly dependent c) Orthogonal d) None of these

Correct Answer: b) Linearly dependent

Explanation: Since g(x) = 2f(x), the functions are scalar multiples of each other, making them linearly dependent.

6. Which set of vectors is linearly independent in ℝ³?

a) {(1,0,0), (0,1,0), (1,1,0)} b) {(1,2,3), (2,4,6)} c) {(1,0,0), (0,1,0), (0,0,1)} d) {(1,1,1), (2,2,2), (3,3,3)}

Correct Answer: c) {(1,0,0), (0,1,0), (0,0,1)}

Explanation: The standard basis vectors are linearly independent as none can be written as a linear combination of the others.

7. The Wronskian of $e x$ and $e 2 x$ is:

0

e^{3 x}

e^{x}

e^{2 x}

Correct Answer: b) $e^{3 x}$

Explanation: Wronskian W = eˣ(2e²ˣ) - e²ˣ(eˣ) = 2e³ˣ - e³ˣ = e³ˣ.

8. The general solution of $y^{''} + 4 y = 0$ is:

y = A \cos 2 x + B \sin 2 x

y = A e^{2 x} + B e^{- 2 x}

y = (A + B x) e^{2 x}

y = A e^{4 x}

Correct Answer: a) $y = A \cos 2 x + B \sin 2 x$

Explanation: Characteristic equation r²+4=0 gives r=±2i, leading to trigonometric solutions.

9. Which of the following is the correct form of particular solution for $y^{''} + 4 y = 3 \cos 2 x$ using undetermined coefficients?

y_{p} = A \cos 2 x

y_{p} = A x \cos 2 x + B x \sin 2 x

y_{p} = A \cos 2 x + B \sin 2 x

y_{p} = A \cos x + B \sin x

Correct Answer: b) $y_{p} = A x \cos 2 x + B x \sin 2 x$

Explanation: Since cos2x matches the homogeneous solution, we multiply by x to get the correct particular solution form.

10. For the vectors v₁ = (1,2,3), v₂ = (4,5,6), and v₃ = (7,8,9), the correct statement about linear dependence is:

a) They are linearly independent b) They are linearly dependent because v₃ = v₁ + v₂ c) They are linearly dependent because det[v₁ v₂ v₃] = 0 d) Both b and c are correct

Correct Answer: d) Both b and c are correct

Explanation: The vectors are dependent since (7,8,9) = (1,2,3)+(4,5,6) and the determinant of the matrix formed by these vectors is zero.

linear differential equation (introduction & solution ) ,linear dependence and its solution

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