The solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 0$$ with boundary conditions<br>$${\text{i}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 0;\,{\text{ii}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 1{\text{ is}}$$

A 60% spirit solution is mixed with r% spirit solution in the ratio 1 : 3 such that a s% spirit solution is obtained. When a (s/3)% spirit solution is mixed with 55% spirit solution in equal ratio, an (r/2)% spirit solution is obtained. Which of the following can be determined? (A) Value of ‘r’ (B) Value of ‘s’ (C) Volume of the 60% spirit solution mixed initially (D) If 9 liters of r% spirit solution was mixed initially, what was the volume of the s% spirit solution thus obtained?

A

Only C and D

B

Only A and B

C

Only A, C and D

D

Only A, B and D

E

Only A, B and C

A wave is first reflected by a boundary P and then by a boundary Q without dissipating energy. If the boundary P is rigid and the boundary Q is non-rigid, then find the fraction of the amplitude of the reflected wave by boundary P to the reflected wave by boundary Q.

A

0

B

∞

C

1 : 1

D

None of these

A chemical solution appears green when the salt concentration is 8 grams per litre (gpl) and red when it is 10 gpl. Pure water was added to the solution of 1 litre of 15 gpl solution till it turned green. Then 125 ml of the green solution is taken out and x grams of salt is added to the green solution to make it red. Another 500 ml of the final solution was taken out and y litres of green solution is added to get a resultant solution of concentration 9.6 gpl. Assume adding salt doesn’t change its volume and find the ratio of x and y.

A

1

B

1.25

C

1.5

D

2

E

1.75

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

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

Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.

A

True

B

False

Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t²
The particular solution of the differential equation is

A

-2 - 2t - t²

B

-2t - t²

C

2t - t²

D

-2 - 2t - 3t²

Consider the differential equation $$\left( {{{\text{t}}^2} - 81} \right)\frac{{{\text{dy}}}}{{{\text{dt}}}} + 5{\text{ty}} = \sin \left( {\text{t}} \right)$$ with y(1) = 2π. Thereexists a unique solution for this differential equation when t belongs to the interval

A

(-2, 2)

B

(-10, 10)

C

(-10, 2)

D

(0, 10)

A differential equation is given as
$${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2{\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{y}} = 4$$
The solution of differential equation in terms of arbitrary constant C₁ and C₂ is

A

$${\text{y}} = \frac{{{{\text{C}}_1}}}{{{{\text{x}}^2}}} + {{\text{C}}_2}{\text{x}} + 2$$

B

$${\text{y}} = {{\text{C}}_1}{{\text{x}}^2} + {{\text{C}}_2}{\text{x}} + 4$$

C

$${\text{y}} = {{\text{C}}_1}{{\text{x}}^2} + {{\text{C}}_2}{\text{x}} + 2$$

D

$${\text{y}} = \frac{{{{\text{C}}_1}}}{{{{\text{x}}^2}}} + {{\text{C}}_2}{\text{x}} + 2$$

Consider the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1 + {{\text{y}}^2}.$$
Which one of the following can be a particular solution of this differential equation?

A

y = tan(x + 3)

B

y = tan x + 3

C

x = tan(y + 3)

D

x = tan y + 3

The differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + 4{\text{y}} = 5$$ is valid in the domain 0 ≤ x ≤ 1 with y(0) = 2.25. The solution of differential equation is

A

y = e^-4x + 1.25

B

y = e^-4x + 5

C

y = e^4x + 5

D

y = e^4x + 1.25

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