Differential Equations

The given differential equation is .

First, we can rewrite the equation in a more standard form:

.

This is a linear first-order differential equation. We can try separating the variables or use an appropriate substitution. After simplifying and solving the equation, we obtain the general solution:

,

where is the constant of integration.

The correct answer is .

The given differential equation is .

We can rewrite this equation as:

,

which is a linear first-order differential equation in the form , where:

and .

The integrating factor is given by:

.

Thus, the integrating factor is (considering ).

The correct answer is .

The given differential equation is .

We can solve this differential equation by using an appropriate substitution or method, such as separating variables or an integrating factor. However, we recognize that this is a first-order non-linear differential equation, and we will attempt a suitable transformation.

By trial and error or using the standard methods for solving such equations, we find that the solution is of the form:

.

The correct answer is .

We are given the differential equation: with the initial condition .
Step 1: Separation of Variables We first separate the variables:
Step 2: Integrate Both Sides Now, integrate both sides: The integral of is and the integral of is , so:
Step 3: Solve for Now, solve for : To simplify the expression, rewrite it as:
Step 4: Use the Initial Condition We are given that . Substituting and into the equation: Solving for :

The correct option is (B) :

We are asked to compute: Step 1: Compute the second derivative of . We start with: The first derivative of is: Now, the second derivative: Step 2: Now, compute : Step 3: Finally, differentiate with respect to :

The correct option is (E) :

We are given the equation: We need to find the value of at .
Step 1: Differentiate implicitly with respect to Differentiate both sides of the equation with respect to .
The derivative of using the product rule:
The derivative of using the product rule:
The derivative of is 0 since it is a constant.
Thus, the differentiated equation is:
Step 2: Solve for Rearrange the equation to isolate : Factor out : Solve for :
Step 3: Substitute into the equation Substitute and into the equation.
First, calculate the terms:
,
,
,
.
Substituting these values into the equation:

So, the correct option is (C) :

The given differential equation is: Rearrange the equation as: We see that this is a first-order linear differential equation in the form: where and . The integrating factor is given by: where is the coefficient of divided by . In this case: Thus, the integrating factor is:

The correct option is (C) : e^4y

We are given the differential equation:
Step 1: Identify the type of equation This is a first-order linear differential equation that can be solved using the method of exact equations or by an appropriate substitution.
Step 2: Check for exactness For the equation to be exact, the following condition must hold: where and .
Derivatives of and :
,
.
Since , the equation is exact.
Step 3: Solve the exact equation Now, we can integrate with respect to and with respect to , and find a solution. The potential function is given by: Integrating with respect to : where is an arbitrary function of . Next, we differentiate with respect to : Equating this with , we get:
Step 4: Solve for Simplifying: Integrating with respect to , we find that is determined up to a constant, and the general solution to the differential equation is:

The correct option is (D) :