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

In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated.

A

False

B

True

The solution of an ODE contains arbitrary constants, the solution to a PDE contains arbitrary functions.

A

True

B

False

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

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

The general solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 2\frac{{{\text{dy}}}}{{{\text{dx}}}} - 5{\text{y}} = 0$$ in terms of arbitrary constants K₁ and K₂ is

A

K1e(-1 + √6)x + K2e(-1 - √6)x

B

K1e(-1 + √8)x + K2e(-1 - √8)x

C

K1e(-2 + √6)x + K2e(-2 - √6)x

D

K1e(-2 + √8)x + K2e(-2 - √8)x

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 difference equation
$${\text{x}}\left( {{\text{ydx}} + {\text{xdy}}} \right)\cos \frac{{\text{y}}}{{\text{x}}} = {\text{y}}\left( {{\text{xdy}} - {\text{ydx}}} \right)\sin \frac{{\text{y}}}{{\text{x}}}$$
Which of the following is the solution of the above equation (c is an arbitrary constant)?

A

$$\frac{{\text{x}}}{{\text{y}}}\cos \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$

B

$$\frac{{\text{x}}}{{\text{y}}}\sin \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$

C

$${\text{xy}}\cos \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$

D

$${\text{xy}}\sin \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$

100 kg of solution A is mixed with 60 kg of solution B . If solution A has tin and copper in the ratio 1: 4 and solution B has lead and tin in the ratio 3 :2 then what is the amount of tin in the new solution?

A

70 kg

B

36 kg

C

44 kg

D

56 kg

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'

Solution X is percent alcohol by volume, and solution Y is 30 percent alcohol by volume. How many milliliters of solution Y must be added to 200 milliliters of solution X to create a solution that is 25 percent alcohol by volume?

A

600

B

800

C

300

D

250

Solution X contains 7% (by weight) of liquid M and solution Y contains 14.5% (by weight) of liquid M. If 3 grams of solution X is mixed with 2 grams of solution Y, then liquid M accounts for what percent of the weight of the resulting solution?

A

6.09%

B

10%

C

10.75%

D

21.5%