Question

1 H grams of artificial sugar in water are being converted into dextrose at a rate which is proportional to the square of the amount unconverted. Find the differential equation expressing the rate of conversion after v minutes given that s grams is converted in v minutes and c being the constant of proportionality.
2 A vehicle of mass m moves along a straight line ( the – axis) while subject to a force indirectly proportional to its displacement x from a fixed point O in its path and 2) a resisting force proportional to its acceleration. Express the total force as a differential equation.
3 Derive the differential equation associated with the primitive
y=Ax^3+Bx^2+Cx+D
where A, B, C and D are arbitrary constants.

Answer 1

Answer on Question #58882 – Math – Differential Equations

Question

1. $H$ grams of artificial sugar in water are being converted into dextrose at a rate which is proportional to the square of the amount unconverted. Find the differential equation expressing the rate of conversion after $v$ minutes given that $s$ grams is converted in $v$ minutes and $c$ being the constant of proportionality.

Solution

The Rate of conversion is given by the following differential equation

$$y' = c(H - y)^2.$$

Initial value:

$$y(v) = s.$$

Here $y$ is the amount of converted sugar,

$$y' = \frac{dy}{dt} \text{ is the rate of conversion}.$$

Answer: $y' = c(H - y)^2, y(v) = s$.

Question

2. A vehicle of mass $m$ moves along a straight line (the – axis) while subject to a force indirectly proportional to its displacement $x$ from a fixed point in its path and a resisting force proportional to its acceleration. Express the total force as a differential equation.

Solution

Total force:

$$F = m a = \frac{k_1}{x} - k_2 a$$

where

$$a = x'' = \frac{d^2 x}{dt^2} \text{ is the acceleration};$$

$k_1, k_2$ are the constants of proportionality.

Thus, the differential equation will be

$$m x ^ {\prime \prime} = \frac {k _ {1}}{x} - k _ {2} x ^ {\prime \prime}.$$

Answer: $mx'' = \frac{k_1}{x} - k_2 x''$ .

Question

3. Derive the differential equation associated with the primitive

$$y = A x ^ {3} + B x ^ {2} + C x + D$$

where $A, B, C$ and $D$ are arbitrary constants.

Solution

Given

$$y = A x ^ {3} + B x ^ {2} + C x + D$$

compute

$$y ^ {\prime} = 3 A x ^ {2} + 2 B x + C,$$ $$y ^ {\prime \prime} = 6 A x + 2 B,$$ $$y ^ {\prime \prime \prime} = 6 A,$$

where $A, B, C$ are arbitrary constants.

From the last equality it follows that a differential equation is

$$\frac {d ^ {4} y}{d x ^ {4}} = 0.$$

Answer: $\frac{d^4y}{dx^4} = 0$ .

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