Attached image contains multiple mathematics questions including solving differential equations, finding Laurent series, testing convergence of series, and working with Fourier ser... Attached image contains multiple mathematics questions including solving differential equations, finding Laurent series, testing convergence of series, and working with Fourier series.
Understand the Problem
The image contains a set of mathematics exam questions, covering various topics such as differential equations, Laurent series, complex analysis, and Fourier series. Each question prompts the student to solve mathematical problems or discuss concepts.
Answer
The solution is \( x(t) = C_1 \cos(\sqrt{3}t) + C_2 \sin(\sqrt{3}t) + \frac{1}{4} e^{t} \).
Answer for screen readers
The complete solution to the differential equation is: $$ x(t) = C_1 \cos(\sqrt{3}t) + C_2 \sin(\sqrt{3}t) + \frac{1}{4} e^{t} $$
Steps to Solve

Identify the problem to solve Select one of the questions to solve. For demonstration, let's use Q2(a): Solve the differential equation ( \frac{d^2x}{dt^2} + 3x = e^{t} ).

Write the characteristic equation The first step for solving a secondorder differential equation with constant coefficients is to find the characteristic equation. For the given ( \frac{d^2x}{dt^2} + 3x = 0 ), we set up: $$ r^2 + 3 = 0 $$ This gives: $$ r = \pm i\sqrt{3} $$

Formulate the general solution of the homogeneous equation The general solution for the homogeneous equation can be expressed as: $$ x_h(t) = C_1 \cos(\sqrt{3}t) + C_2 \sin(\sqrt{3}t) $$

Find a particular solution To find the particular solution ( x_p(t) ) for the nonhomogeneous part ( e^{t} ), we use the method of undetermined coefficients. Assume: $$ x_p(t) = A e^{t} $$ Take the first and second derivatives: $$ x_p'(t) = A e^{t} $$ $$ x_p''(t) = A e^{t} $$

Substitute back into the equation Plug the particular solution into the original equation: $$ A e^{t} + 3A e^{t} = e^{t} $$ Thus: $$ 4A e^{t} = e^{t} $$ This implies: $$ 4A = 1 \Rightarrow A = \frac{1}{4} $$

Combine general and particular solutions The complete solution is given by: $$ x(t) = x_h(t) + x_p(t) $$ Thus: $$ x(t) = C_1 \cos(\sqrt{3}t) + C_2 \sin(\sqrt{3}t) + \frac{1}{4} e^{t} $$
The complete solution to the differential equation is: $$ x(t) = C_1 \cos(\sqrt{3}t) + C_2 \sin(\sqrt{3}t) + \frac{1}{4} e^{t} $$
More Information
This solution includes both the homogeneous and particular parts. The constants ( C_1 ) and ( C_2 ) are determined by initial conditions, which are not provided in the problem statement.
Tips
 Forgetting to include the homogeneous solution when finding the total solution.
 Misapplying the method of undetermined coefficients by using the wrong form of the particular solution.