Differential Equations

Step 1: Finding the derivative.
The function is a rational function, and its derivative is given by applying the quotient rule or recognizing it as a standard function. The derivative is , which has the shape of a curve that is symmetric about the origin and decreases as increases.
Step 2: Conclusion.
Thus, the correct answer is option (B), as the graph of matches the expected shape of the derivative.

Step 1: Solve the differential equation.
We start with the second-order linear differential equation: This is a standard form of a differential equation with constant coefficients. The characteristic equation is: Solving for , we find: Thus, the general solution to the differential equation is: Step 2: Apply initial conditions.
Using the initial conditions and , we substitute into the general solution and its derivative: Substituting : Step 3: Final solution.
Thus, the solution is: Step 4: Find .
Substituting : Step 5: Conclusion.
Thus, the value of is approximately 0.2707.

Step 1: Set up the differential equations.
We are given the following system of differential equations: These equations are coupled and linear. We can solve them by taking the second derivative of both equations and eliminating and .
Step 2: Solve the system.
Taking the derivative of with respect to gives: This simplifies to: Now, solving this system gives a solution of the form , where is the constant we need to determine.
Step 3: Conclusion.
The value of is 1.

Step 1: Solve the differential equation.
The equation is a first-order linear DE. Using the integrating factor , the solution is

Step 2: Apply the initial condition.
, which agrees.

Step 3: Analyze behaviour of solution.
As , Thus the curve starts at and asymptotically approaches .

Step 4: Match with the options.
Only option (B) shows a saturation at .