Advanced Engineering Mathematics — Summer 2024

Solve 9yy′ +4x = 0.

Solve the initial value problem y′ −(1+3x−1)y = x+2;y(1) = e−1.

Find the Fourier series of the function f(x) = x2;−𝜋 < x < 𝜋.

Find the general solution of y′′ +3y′ +2y = 0.

Solve y′′′ −3y′′ +3y′ −y = 4et.

Using the method of variation of parameters find the general solution of (D2 − 2D + 1)y = 3x2ex.

Using the method of undermined coefficients, find a particular solution of y′′ −4y′ −12y = 8x2.

Obtain the Fourier series for the function f(x) given by 2x 1+( );−𝜋 ≤ x ≤ 0 f(x) = { 𝜋 . 2x 1−( );0 ≤ x ≤ 𝜋 𝜋 1 1 1 𝜋2 Hence, deduce that + + + ⋯ = .

1;−1 ≤ x ≤ 1 Afunction f(x) is defined by f(x) = { 0;otℎerwise Find the Fourier integral representation of f(x). ∞sincosx ∞sin Hence, evaluate (a) ∫ d. (b) ∫ d. 0  0 

Find a cosine series of period 2𝜋 to represent f(x) = sinx in 0 < x < 𝜋. Also, graph the corresponding periodic continuation of f(x). Hence 1 1 1 𝜋 deduce that 1 − + − + ⋯.=

Determine the series solution for the differential equation y′′ +y = 0 about x = 0.0

3 Find the Laplace transform of t3 + e−3t + t2 .

Find the Laplace transform of cosℎ(kt)coskt

2s+3 Find the inverse Laplace transform of (s+2)(s+1)2

Define (i) Gamma function and Beta function (ii) Write the relation between Beta and Gamma function.

0;0 ≤ t < k Find the Laplace transform of unit step function f(t) = { 1; t ≥ k

Solve the IVP using the Laplace transform: y′′ +4y = 0;y(0) = 1,y′(0) = 6.

Form the partial differential equation by eliminating the arbitrary constants for az+b = a2x +y.

Solve (y+z)p−(x+z)q = x −y.

𝜕u 𝜕u Using the method of separation of variables, solve = 2 + u. 𝜕x 𝜕t

Find the complete integral of z = px+qy+pq.

Solve (D2 +10DD′ +25D′2 )z = e3x+2y.

The base of semi-infinite strip of metal plate is 30cm and is kept at 100°C. The two long edges are at zero temperature. Find the temperature at any point 15cm away from the base and situated midway between the long edges.