Define: Dirac Delta function, Gamma function, Laplace Transform of a function.
[3 marks]Solve (D2 −4)y = 1+ex ; where D = d/dx.
[4 marks]𝜋+x ; −𝜋 < x < 0 Find the Fourier series for f(x) = { 𝜋−x ; 0 < x < 𝜋
[7 marks]d4y d2y Solve −2 +y = 0. dx4 dx2
[3 marks]Solve Sinℎx Cosy dx = Cosℎx Siny dy.
[4 marks]Find the Half range Cosine series for f(x) = (x−1)2 in (0,1).
[7 marks]Show that current in a circuit containing Resistance R, Inductance Land Constant emf Eis given by i = E [1−e − R Lt]. R
[7 marks]Solve x2y′′ +xy′ +y = 0.
[3 marks]Solve by the method of undetermined coefficients. y′′ +10y′ +25y = e−5x
[4 marks]Solve using method of variation of parameters. y′′ +2y′ +y = e−xCosx
[7 marks]State and prove First shifting theorem of Laplace Transform.
[3 marks]Sin x ;0 ≪ x ≪ 𝜋 Express f(x) = { 0 ; x > 𝜋 ∞Sin 𝜆x Sin 𝜋𝜆 as Fourier Sine integral and evaluate ∫ d𝜆 0 1−𝜆2
[4 marks]Solve in series the equation y′ = 3x2y.
[7 marks]Find L[t Sint]
[3 marks]Find L−1[ 4s+5 ] (s−1)2(s+2)
[4 marks]State Convolution theorem and hence find L−1[ s ] (s2+a2)2
[7 marks]Define unit step function u(t−a). Find L[t2u(t−2)].
[3 marks]Solve the differential equation. (D3 −2D2 +4D −8)y = 0 ; where D = d/dx1
[4 marks]Solve differential equation using Laplace transform. y′′ +2y′ +y = e−t ; y(0) = −1,y′(0) = 1
[7 marks]Find Radius of convergence of the power series. ∞ xn ∑ n!0
[3 marks]Solve the partial differential equation. 𝜕z 𝜕z p+q = z ;wℎere p = and q = 𝜕x 𝜕y
[4 marks]Prove that Laplace Equation in polar coordinates is 𝜕2u 1𝜕u 1 𝜕2u ∇2u = + + = 0 𝜕r2 r 𝜕r r2𝜕𝜃2
[7 marks]Form partial differential equation by eliminating arbitrary functions. f(xy+z2,x+y+z) = 0
[3 marks]Solve partial differential equation. 𝜕2z +z = 0, given that when x = 0;z = ey and 𝜕z = 1. 𝜕x2 𝜕x
[4 marks]Solve the partial differential equation using method of separation of variables. 𝜕u 𝜕u = 4 ; u(0,y) = 8e−3y 𝜕x 𝜕y
[7 marks]