Solve yy′ −2x = 0.
[3 marks]Find L−1[ 2s+1 ]. s(s+1)
[4 marks]Expand f(x) = 1 (𝜋−x) as a Fourier series in the interval (0,2𝜋).2
[7 marks]Define unit step function and rectangle function.
[3 marks]d2y dy Solve + 5 + 6y = 12ex . dx2 dx
[4 marks]Find a power series solution of y′′ +xy = 0 about the ordinary point x = 0.
[7 marks]Find the Fourier series of 0, −𝜋 < x < 0 f(x) = { x, 0 < x < 𝜋
[7 marks]t t Find L[∫ ∫ sinat dt dt].00
[3 marks]Solve (D2 +9)y = sin2x+cos4x.
[4 marks]1 Find the inverse Laplace transform of by Convolution theorem. (s2+4)2
[7 marks]dy Solve +2ytanx = sinx. dx
[3 marks]Find L−1[ 6+s ]. s2+6s+13
[4 marks]Solve y′′ +4y = 8x2 by method of undetermined coefficients.
[7 marks]Find the half range sine series of f(x) = ex in 0 < x < 𝜋.
[3 marks]Find the Laplace transform of 1) te−t 2) e−atcosbt.
[4 marks]Solve y′′ +y = cosecx by method of variation of parameters.
[7 marks]sin2t Find the Laplace transform of . t
[3 marks]Solve (D2 +10DD′ +25D′2 )z = e3x+2y.
[4 marks]Solve by Laplace Transform y′ +2y = e−3t with y(0) = 1.
[7 marks]Form a partial differential equation for the equation z = (x−3)2 + (y−4)2.1
[3 marks]Solve p−x2 = q +y2.
[4 marks]Solve z = pq by Charpit’s method.
[7 marks]Solve z = px+qy−2√pq.
[3 marks]Solve xp+yq = 3z.
[4 marks]𝜕u 𝜕u Solve = 4 , given that u(0,y) = 8e−3y by the method of 𝜕x 𝜕y separation of variables.
[7 marks]