What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)2 + 1 = 0?

The figure shows the plot of y as a function of x
Differential Equations mcq question image

The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is

A

$$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 1$$

B

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{x}}$$

C

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - {\text{x}}$$

D

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$

Children are in pursuit of a dog whose leash has broken. James is directly behind the dog. Ruby is behind James. Rachel is behind Ruby. Max is ahead of the dog walking down the street in the opposite direction. As the children and dog pass, Max turns around and joins the pursuit. He runs in behind Ruby. James runs faster and is alongside the dog on the left. Ruby runs faster and is alongside the dog on the right. Which child is directly behind the dog?

A

James

B

Ruby

C

Rachel

D

Max

Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R). Read the statements and choose the correct answer:
Assertion (A) The nominal interest rate comprises of a real interest rate and an expected rate of inflation and it adjusts when the inflation rate is expected to change. Hence, in the perfect international capital markets, real rate of returns are equal in the two countries.
Reason (R) The international fisher effects states that the nominal interest rate differential must be equal to the expected inflation rate differential In the two countries.

A

Both (A) and (R) are correct and (R) is the right explanation of (A)

B

Both (A) and (R) are correct, but (R) is not the right explanation of (A)

C

(A) is correct, but (R) is not correct

D

Both (A) and (R) are incorrect

Solution of a differential equation is any function which satisfies the equation.

A

False

B

True

Consider with respect to HusserI's concept of phenomena.
1. Phenomena is a substantial unity that has real properties, real parts and real changes.
2. Phenomena is no substantial unity, that has no real parts and no real changes.
3. Phenomena has no nature, but only essence.
4. Phenomena have no essence, but only nature.

A

Only (1) is true

B

Only (2) and (3) are true

C

Only (1) and (3) are true

D

Only (1), (3) and (4) are true

A function n(x) satisfies the differential equation $$\frac{{{{\text{d}}^2}{\text{n}}\left( {\text{x}} \right)}}{{{\text{d}}{{\text{x}}^2}}} - \frac{{{\text{n}}\left( {\text{x}} \right)}}{{{{\text{L}}^2}}} = 0$$ where L is a constant. The boundary conditions are: n(0) = K and n($$\infty $$) = 0. The solution to this equation is

A

n(x) = K exp(x/L)

B

n(x)= K exp(-x/√L)

C

n(x) = K² exp(-x/L)

D

n(x) = K exp(-x/L)

Under the Evidence Act facts not otherwise relevant are relevant . . . . . . . .
1. If they are significant
2. If they are inconsistent with any fact in issue or relevant fact
3. If they make the existence or nonexistence of any fact in issue or relevant fact highly probable or improbable
Choose correct answer:

A

1, 2 and 3

B

2 only

C

1 and 2 only

D

2 and 3 only

A curve passes through the point (x = 1, y = 0) and satisfies the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{{{\text{x}}^2} + {{\text{y}}^2}}}{{2{\text{y}}}} + \frac{{\text{y}}}{{\text{x}}}.$$ The equation that describes the curve is

A

$$\ln \left( {1 + \frac{{{{\text{y}}^2}}}{{{{\text{x}}^2}}}} \right) = {\text{x}} - 1$$

B

$$\frac{1}{2}\ln \left( {1 + \frac{{{{\text{y}}^2}}}{{{{\text{x}}^2}}}} \right) = {\text{x}} - 1$$

C

$$\ln \left( {1 + \frac{{\text{y}}}{{\text{x}}}} \right) = {\text{x}} - 1$$

D

$$\frac{1}{2}\ln \left( {1 + \frac{{\text{y}}}{{\text{x}}}} \right) = {\text{x}} - 1$$

The function f(t) satisfies the differential equation $$\frac{{{{\text{d}}^2}{\text{f}}}}{{{\text{d}}{{\text{t}}^2}}} + {\text{f}} = 0$$ and the auxiliary conditions, f(0) = 0, $$\frac{{{\text{df}}}}{{{\text{dt}}}}\left( 0 \right) = 4.$$ The Laplace transform of f(t) is given by

A

$$\frac{2}{{{\text{s}} + 1}}$$

B

$$\frac{4}{{{\text{s}} + 1}}$$

C

$$\frac{4}{{{{\text{s}}^2} + 1}}$$

D

$$\frac{2}{{{{\text{s}}^2} + 1}}$$

A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x² - 2y² + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

A

4y² - 4xy + constant

B

-4xy + 2y² - 2x² + constant

C

2x² - 2y² + xy + constant

D

4xy - 2x² + 2y² + constant

What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)2 + 1 = 0?

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