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Consider the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1 + {{\text{y}}^2}.$$<br>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
Correct Answer:
y = tan(x + 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
Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t
2
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>
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
$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$
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 × 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>
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)
Suppose you have four test tubes labelled as 'A', 'B', 'C' and 'D'. 'A' contains plain water, 'B' contains solution of an alkali, 'C' contains solution of an acid, and 'D' contains solution of sodium chloride. Which one of these solution will turn phenophthalein solution pink?
A
Solution 'A'
B
Solution 'B'
C
Solution 'C'
D
Solution 'D'
A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.
A
True
B
False
