Differential Equations

{Step 1:} Rewrite the Differential Equation\\ We start with the given differential equation: To simplify, we exponentiate both sides to eliminate the logarithm:

{Step 2: Separate the Variables} The equation is separable, meaning we can rearrange terms to group all -terms on one side and -terms on the other:

{Step 3: Integrate Both Sides} We integrate both sides to find the general solution: where is the constant of integration.

{Step 4: Solve for } Multiply both sides by to simplify: Let , then: Taking the natural logarithm of both sides: This expression represents the general solution of the differential equation, containing one arbitrary constant .

{Step 5: Apply the Initial Condition} We are given the initial condition . Substitute and into the general solution to find : Multiply both sides by : Exponentiate both sides: Substitute back into the general solution:

This is the particular solution satisfying the initial condition .

{Step 6: Determine the Number of Arbitrary Constants} In the general solution, there was one arbitrary constant . However, after applying the initial condition, this constant was determined uniquely.

Therefore, the particular solution has: arbitrary constants.

Step 1: {Rewriting the equation}
Divide through by :

Step 2: {Find the integrating factor}
The integrating factor is determined by identifying the dependency on and multiplying the equation by .

Step 3: {Verify integrating factor}
After multiplying, the left-hand side becomes exact. The integrating factor is , which matches option (D).

To determine the order and degree of the differential equation, follow these steps:

1. Order: The order of a differential equation is the highest order derivative present in the equation. In the given equation: the highest derivative is .

Thus, the order of the equation is . 2. Degree: The degree of a differential equation is defined as the power of the highest order derivative, provided the equation is free from radicals and fractional powers of the derivatives.

In this case, appears to the first power, and there are no fractional powers of in the equation.

Thus, the degree of the equation is . Hence, the order and degree of the given differential equation are and , respectively, and the correct answer is (C).

Step 1: Separating the Variables Rewriting the equation:

Step 2: Integrating Both Sides Integrate both sides: The left-hand side becomes: The right-hand side uses the integral of : So the equation becomes: Here, is the constant of integration.

Step 3: Simplify the Expression} Combine the logarithms: Exponentiate both sides to remove the logarithm:

Step 4: Finding the Particular Solution} We are given the condition . Substitute and into the solution: Simplify: Since , we have:

Step 5: Final Solution} Substitute back into the solution:

Final Answer: This is the required particular solution to the given differential equation.

To solve the problem, we need to determine the order and degree of the given differential equation:

1. Simplify the Differential Equation:
Let us first simplify the expression:
Let , then the equation becomes:

That implies is constant, so:
, where is constant

We were given:

We need to look at the highest order derivative involved in the equation as written — the derivative is acting on a power of , making the expression effectively include a second derivative when expanded.

Differentiating:
So, the highest derivative is , which means:

Order (p) = 2
The equation is polynomial in its highest order derivative (linear in ), so:
Degree (q) = 1

2. Final Calculation:

Final Answer:
The value of is 1.