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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
-2 - 2t - t<sup>2</sup>
B
-2t - t<sup>2</sup>
C
2t - t<sup>2</sup>
D
-2 - 2t - 3t<sup>2</sup>
Correct Answer:
-2 - 2t - t<sup>2</sup>
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
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 figure shows the plot of y as a function of x
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
$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$
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 × 10<sup>5</sup>
B
6.12 × 10<sup>5</sup>
C
7.24 × 10<sup>5</sup>
D
9.08 × 10<sup>5</sup>
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 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<sup>x</sup>
B
x<sup>2</sup>
C
$$\frac{1}{{\text{x}}}$$
D
$$l$$n x
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 $$\left( {{{\text{t}}^2} - 81} \right)\frac{{{\text{dy}}}}{{{\text{dt}}}} + 5{\text{ty}} = \sin \left( {\text{t}} \right)$$ 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 particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
A
True
B
False
Consider the following statements about emulsion SBR and solution SBR. I. Solution SBR has higher content of non-rubber materials than emulsion SBR II. Solution SBR has higher molecular weight but broader molecular weight distribution than emulsion SBR III. Solution SBR has higher cis-1,4-polybutadiene content than emulsion SBR Which of the following statements are true?
A
I, II, III
B
I, II
C
III only
D
I only