How to solve QUESTION 1 (30 marks) a ) Determine the singular points of the differential equation (x^ 2 -1)y^ prime prime -2xy^ prime +4y=0 and find the nature of their singularity (5marks) b) For which values of x does the series Sigma n=1 ^ infty x^ k k^ 2 converge?(5marks) ) Evaluate integrate x ^ 2 * e ^ (-2x ^ 2) dx from 0 to ∞ (5marks) d) Solve- partial u partial t =2tcos theta given that u = 2t when theta = 0 (5marks) e)Find the inverse Laplace transform of (2s ^ 2 - 6s - 1)/((s - 3)(s ^ 2 - 2s + 5)) (5marks) f) Determine the coefficients a_{0i}*a_{n} and b_{n} of the Fourier series to represent the periodic function defined as f(x) = x/2; 0

Q: How to solve QUESTION 1 (30 marks) a ) Determine the singular points of the differential equation (x^ 2 -1)y^ prime prime -2xy^ prime +4y=0 and find the nature of their singularity (5marks) b) For which values of x does the series Sigma n=1 ^ infty x^ k k^ 2 converge?(5marks) ) Evaluate integrate x ^ 2 * e ^ (-2x ^ 2) dx from 0 to ∞ (5marks) d) Solve- partial u partial t =2tcos theta given that u = 2t when theta = 0 (5marks) e)Find the inverse Laplace transform of (2s ^ 2 - 6s - 1)/((s - 3)(s ^ 2 - 2s + 5)) (5marks) f) Determine the coefficients a_{0i}*a_{n} and b_{n} of the Fourier series to represent the periodic function defined as f(x) = x/2; 0 <= x <= 2pi; f(x) = f(x + 2pi) ie the period is 2pi (5marks)

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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How to solve QUESTION 1 (30 marks) a ) Determine the singular points of the differential equation (x^ 2 -1)y^ prime prime -2xy^ prime +4y=0 and find the nature of their singularity (5marks) b) For which values of x does the series Sigma n=1 ^ infty x^ k k^ 2 converge?(5marks) ) Evaluate integrate x ^ 2 * e ^ (-2x ^ 2) dx from 0 to ∞ (5marks) d) Solve- partial u partial t =2tcos theta given that u = 2t when theta = 0 (5marks) e)Find the inverse Laplace transform of (2s ^ 2 - 6s - 1)/((s - 3)(s ^ 2 - 2s + 5)) (5marks) f) Determine the coefficients a_{0i}*a_{n} and b_{n} of the Fourier series to represent the periodic function defined as f(x) = x/2; 0 <= x <= 2pi; f(x) = f(x + 2pi) ie the period is 2pi (5marks)

Accepted Answer

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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