Ordinary Differential Equation T 2

Rsquared Mathematics

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Time: 45:00

Topic: Ordinary Differential Equation

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B.

1. If \(y(x)\) be the solution of the equation \(\cos x \, dy = (\sin x - y) \, dx\) then

(a.) \(\lim_{x \rightarrow 0} y(x) > 1\)
(b.) \(\lim_{x \rightarrow \pi} y(x) < 0\)
(c.) \(\lim_{x \rightarrow \pi} y(x) > 0\)
(d.) None of these

Explanation: \( \cos x \, dy = (\sin x - y) \, dx \)
\( \Rightarrow y' - y \tan x = -\sec x \cdot y^2 \)
\( \therefore \frac{dz}{dx} + z \tan x = \sec x \Rightarrow z \sec x = \tan x + c \)

2. Consider the differential equation \((x+2)^2(x-1) \, y^{\prime \prime} + 3(x-1) \, y^{\prime} + 2y = 0\) then

(a.) \(x=-2\) is the only singular point
(b.) \(x=-2 \& 1\) are singular points
(c.) \(x=1\) is the only singular point
(d.) \(x=1\) is not a singular point

Explanation: Both -2 and 1 are singular points of the ode.

3. Which of the following are LI pair of solutions of the DE \(y^{\prime \prime} + p(x) \, y^{\prime} + q(x) \, y = 0\) on \([-1,1]\) where \(p(x)\) and \(q(x)\) are continuous functions on \([-1,1]\).

(a.) \(x \sin x, \cos x\)
(b.) \(x e^x, \sin x\)
(c.) \(x^2, \cos x\)
(d.) \(e^{x-1}, e^x-1\)

Explanation: Wronskian is zero for a, b, and c.

4. If \(y(x) = c_1 e^{ax} + c_2 e^{\rho_x} + c_3 e^{y x}\) is a general solution of \(\frac{d^3 y}{d x^3} - p \frac{d^2 y}{d x^2} + q \frac{d y}{d x} - r y = 0\) and \(e^{\alpha x} \cdot e^{\theta_x} = 1\) or \(e^{\alpha x} \cdot e^{\gamma x} = 1\) or \(e^{\rho_X} \cdot e^{\gamma x} = 1\), where \(p, q\), and \(r\) are constants. Then which of the following is true -

(a.) \(pq = r\)
(b.) \(pr = q\)
(c.) \(pq = \frac{1}{r}\)
(d.) \(pq = r^2\)

Explanation: Notice that the sum of the roots is \(p\) and \(\alpha, \beta, \gamma\) are roots of the auxiliary equation of the given DE.

5. The differential equation \(\frac{d^2 y}{d x^2} + 2(\alpha-1)(\alpha-3) \frac{d y}{d x} + (\alpha-2) y = 0\) has a solution \(y(x) = a \cos (\beta x) + b \sin (\beta x)\) for some non-zero real \(a, b, \beta\), then the value of \(\alpha\)

(a.) 1
(b.) 2
(c.) 3
(d.) 4

Explanation: Hit and try / try to find the root of the auxiliary equation of DE.

Part - C

6. Consider the boundary value problem \(-y^{\prime \prime}=f(x)\) for \(0 < x < 1, y^{\prime}(0)=y^{\prime \prime}(0)=0\). In which of the following cases does there exist a solution to this problem -

(a.) \(f(x) = \cos \pi x\)
(b.) \(f(x) = x - \frac{1}{2}\)
(c.) \(f(x) = \sin \pi x\)
(d.) None of these

Explanation: Use hit and try method to determine solutions.

7. \(y^{\prime \prime} + \left(e^{\sin x} + x^2 + \cos^{-1} x\right) y^{\prime} + y \cdot e^{\tan^{-1} x} = \sin x\), if \(y_1 \& y_2\) are solutions then which of the following are also solutions -

(a.) \(\frac{1}{3}\left(4 y_1 - y_2\right)\)
(b.) \(\frac{1}{2}(y_1 + y_2) \)
(c.) \(2 y_1 - y_2) \)
(d.) \(\frac{1}{2}(6 y_1 - 4 y_2 ) \)

Explanation: Linear combination of solutions of non-homogeneous DE is a solution if some coefficient is 1.

8. Let \(y_1=e^x\) and \(y_2=xe^x\) be solutions of the differential equation \(y^n - 2y + y = 0\) and \(\pi(x)\) defines the Wronskian of \(y_1 \& y_2\), then choose the incorrect statement -

(a.) \(W(x) \neq 0\) for all \(x \in \mathbb{R}\)
(b.) \(y_1 \& y_2\) form a fundamental set of solutions for all \(x \in \mathbb{R}\)
(c.) \(y_1 \& y_2\) are linearly dependent for all \(x \in \mathbb{R}\)
(d.) \(W(x)\) is a continuously differentiable function

Explanation: Solutions are linearly dependent on \(\mathbb{R}\).

9. The equation of the tangent to the curve \(y=x+\frac{4}{x^2}\) that is parallel to the \(x\)-axis is -

(a.) \(y=1\)
(b.) \(y=2\)
(c.) \(y=3\)
(d.) \(y=0\)

Explanation: The slope \(\frac{dy}{dx} = 0\) implies \(1 - \frac{8}{x^3} = 0\), so \(x = 2\) and \(y = 3\).

10. The differential equation of the family of circles with a fixed radius of 5 units \& center on the line \(y=2\) is -

(a.) \((y-2)^2\left(\frac{dy}{dx}\right)^2=25-(y-2)^2\)
(b.) \((y-2)^2\left(\frac{dy}{dx}\right)^2=25-(y+2)^2\)
(c.) \((x-2)\left(\frac{dy}{dx}\right)^2=25-(y-2)^2\)
(d.) \((x-2)^2\left(\frac{dy}{dx}\right)^2=25-(y+2)^2\)

Explanation: \( \begin{aligned} & \text{Exp: } (x-h)^2 + (y-2)^2 = 5^2 \\ & 2(x-h) + 2(y-2) \frac{dy}{dx} = 0 \\ & (x-h) + (y-2) \frac{dy}{dx} = 0 \\ & (x-h) = -(y-2) \frac{dy}{dx} \\ & (y-2)^2 \left(\frac{dy}{dx}\right)^2 + (y-2)^2 = 5^2 \\ & (y-2)^2 \left(\frac{dy}{dx}\right)^2 = 25 - (y-2)^2 \end{aligned} \)