Differential Equations

Separate variables: Integrate: Apply condition , : Answer: .

Rewrite: Integrate: Solution: .

Rewrite the equation: Since , the equation becomes: The degree of a differential equation is the power of the highest-order derivative when the equation is polynomial in derivatives. Here, the highest-order derivative is , with power 1. Thus, the degree is:

Separate variables: .
Partial fractions: . Solve: Options seem incorrect; solution is complex.

Rearrange the terms: Divide both sides by and : Integrating both sides: The integral of is , and the integral of is . Thus, we get: Exponentiate both sides to remove the logarithms: Finally, we can solve for , but this equation gives the general solution to the differential equation.

Step 1: Solve the differential equation.
The given equation can be solved using separation of variables, yielding the solution . Step 2: Conclusion.
Thus, the correct answer is (A).

The given differential equation is: The degree of a differential equation is the highest power of the highest order derivative after making the equation free from derivatives in fractions or irrational powers. Here, the highest order derivative is , and its power is 1. Thus, the degree of the differential equation is:

Step 1: Let , where is a constant. Step 2: First, compute the first derivative of : Step 3: Compute the second derivative: Step 4: Substitute into the equation : Thus, we have shown that:

Step 1: Separate the variables in the given differential equation: Rearrange to separate the variables: Step 2: Integrate both sides: The left side: The right side: So, we have: Step 3: Apply the initial condition : Step 4: Substitute into the equation: Step 5: Take the natural logarithm: Thus, the particular solution is:

Step 1: Substituting
Step 2: Rewrite the Equation
Step 3: Solve the Differential Equation

Step 1: Relating Acceleration to Velocity
Given that acceleration is proportional to velocity , we have: Since acceleration is the rate of change of velocity, we can write: Step 2: Formulating the Differential Equation
Rearrange the equation to separate the variables: This is the desired first-order differential equation.

Step 1: Use Substitution
Let , so that: Step 2: Rewrite the Equation
Step 3: Solve the Separable Equation
Step 4: Integrate Both Sides
Step 5: Substitute Back
Taking exponentials on both sides:

Step 1: First Derivative
Since , the first derivative of with respect to is: Step 2: Second Derivative
To find the second derivative, differentiate with respect to : Using the chain rule: Step 3: Verify the Given Identity
We need to show that: Substitute the expressions for and : Simplify the first term: Simplify the second term: Now, combine the terms: Factor out : Simplify the expression inside the parentheses: This simplifies to: Thus, we have shown that:

Step 1: Rewriting the Equation in Standard Form
The given equation: can be rewritten as: Step 2: Determining the Integrating Factor (IF)
The integrating factor is computed as: