Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t2The particular solution of the differential equation is

Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t<sup>2</sup><br>The particular solution of the differential equation 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 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

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

Consider the Assertion (A) and Reason (R) and select the correct answer:
Assertion (A) If one premise is particular, the conclusion must be particular.
Reason (R) (i) An affirmative particular has no distributed terms, and a negative particular has an only one.
(ii) The premises cannot both be particular and thus must differ in quantity.

A

A and R both are true but R (i) and (ii) correct explanations of A

B

A and R both are true but R (i) is correct explanation of A

C

A and R both are true but R (ii) is a correct explanation of A

D

A is true, but R (i) and (ii) are incorrect explanations of A

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

It is desired to concentrate a 20% salt solution (20 kg of salt in 100 kg of solution) to a 30% salt solution in an evaporator. Consider a feed of 300 kg/min at 30°C. The boiling point of the solution is 110°C, the latent heat of vaporisation is 2100 kJ/kg and the specific heat of the solution is 4 kJ/(kgK). The rate at which the heat has to be supplied in (kJ/min) to the evaporator is

A

3.06 × 10⁵

B

6.12 × 10⁵

C

7.24 × 10⁵

D

9.08 × 10⁵

What is the general form of the general solution of a non-homogeneous DE (uh(t)= general solution of the homogeneous equation, up(t)= any particular solution of the non-homogeneous equation)?

A

u(t)=uh (t)/up (t)

B

u(t)=uh (t)*up (t)

C

u(t)=uh (t)+up (t)

D

u(t)=uh (t)-up (t)

Consider the differential equation $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} - {\text{y}} = 0.$$ Which of the following is a solution to this differential equation for x > 0 ?

A

e^x

B

x²

C

$$\frac{1}{{\text{x}}}$$

D

$$l$$n x

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)

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

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