Differential Calculus

Step 1: Write the differential equation clearly
The given differential equation is:
y dx - x dy + (x log x) dx = 0

Step 2: Rearrange terms
Group dx and dy terms:
(y + x log x) dx - x dy = 0

Step 3: Express in differential form
Rewrite as:
(y + x log x) dx = x dy

Step 4: Divide both sides by x dx

or
dy = dx

Step 5: Recognize this as a linear differential equation
It can be written as:

Step 6: Find the integrating factor (IF)
IF =

Step 7: Multiply through by the integrating factor

This simplifies to:

Step 8: Integrate both sides

Step 9: Evaluate the integral
Let
Use substitution , so
Therefore,

Step 10: Write the general solution

Multiplying both sides by x:

Final Answer:

Step 1: Understand the differential equation
The given differential equation is dy/dx + y = 0, which is a first-order linear differential equation.

Step 2: General form of a first-order linear differential equation
A first-order linear differential equation can be written as dy/dx + P(x)y = Q(x). In this case, P(x) = 1 and Q(x) = 0.

Step 3: Solve the differential equation
To solve, use the integrating factor method.
The integrating factor (IF) = e^(∫P(x) dx) = e^(∫1 dx) = e^x.

Step 4: Multiply the entire equation by the integrating factor
e^x * dy/dx + e^x * y = 0
This can be written as d/dx (y * e^x) = 0.

Step 5: Integrate both sides
∫ d/dx (y * e^x) dx = ∫ 0 dx
y * e^x = C, where C is the constant of integration.

Step 6: Express the general solution
y = C * e^(-x).

Step 7: Number of arbitrary constants
The general solution contains one arbitrary constant C.

Final Answer: (B) 1

The function can be written as:

To check differentiability at , we compute the left-hand derivative (LHD) and the right-hand derivative (RHD).

1. Right-hand derivative (RHD): At , RHD:

2. Left-hand derivative (LHD): At , LHD:

Since , the derivative exists and is continuous. Therefore, is differentiable at .

Step 1: Volume of a Cylinder Formula

The volume of a cylinder is given by:

Where:

• (radius)
• = height of the sugar in the tank (variable)
• = volume of the sugar in the tank (function of time)

Step 2: Differentiate Both Sides with Respect to Time

Differentiate both sides with respect to :

Step 3: Substitute Known Values

Given:

Substitute into the formula:

Divide both sides by :

Step 4: Conclusion

The correct answer is: (A) 0.1 cm/s