2018-2019 December

Question no 1 compulsory

Attempt ant tree Questions from remaining five questions.

Evaluate `int_0^inftye^(x^3)/sqrtx dx`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Find the length of the curve `x=y^3/3+1/(4y)` from `y=1 to y=2`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(D^2+2)y=e^xcosx+x^2e^(3x)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Evaluate `int_0^1 int_0^(x2) y/(ex) dy dx`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(4x+3y-4)dx+(3x-7y-3)dy=0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `dy/dx=1+xy` with initial condition `x_0=0,y_0=0.2` By Taylors series method. Find the approximate value of y for x= 0.4(step size = 0.4).

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(d^2y)/dx^2-16y=x^2 e^(3x)+e^(2x)-cos3x+2^x`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Show that `int_0^pi log(1+acos x)/cos x dx=pi sin^-1 a 0 ≤ a ≤1.`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Change the order of integration and evaluate `int_0^2 int_(2-sqrt(4-y^2))^(2+sqrt(4-y^2)) dxdy`

Concept: Application of Double Integrals to Compute Volume

Evaluate `int int int (x+y+z)` `dxdydz ` over the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1.

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Find the mass of lamina bounded by the curves 𝒚 = 𝒙𝟐 − 𝟑𝒙 and 𝒚 = 𝟐𝒙 if the density of the lamina at any point is given by `24/25 xy`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `x^2 (d^2y)/dx^2+3x dy/dx+3y =(log x.cos (log x))/x`

Concept: Equations Reducible to Exact Form by Using Integrating Factors

Find by double integration the area bounded by the parabola 𝒚𝟐=𝟒𝒙 And 𝒚=𝟐𝒙−𝟒

Concept: Application of Double Integrals to Compute Area

Solve `dy/dx+x sin 2 y=x^3 cos^2 y`

Concept: Runga‐Kutta Fourth Order Formula

Solve `dy/dx=x^3+y`with initial conditions y(0)=2 at x= 0.2 in step of h = 0.1 by Runge Kutta method of Fourth order.

Concept: Runga‐Kutta Fourth Order Formula

Evaluate `int_0^1 x^5 sin ^-1 x dx`and find the value of β `(9/2,1/2)`

Concept: Exact Differential Equations

In a circuit containing inductance L, resistance R, and voltage E, the current i is given by `L (di)/dt+Ri=E`.Find the current i at time t at t = 0 and i = 0 and L, R and E are constants.

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Evaluate `int_0^6 dx/(1+3x)`by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule.

Concept: Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem

Find the volume bounded by the paraboloid 𝒙^{𝟐}+𝒚^{𝟐}=𝒂𝒛 and the cylinder 𝒙^{𝟐}+𝒚^{𝟐}=𝒂^{𝟐. }

Concept: Application of Double Integrals to Compute Mass

Change to polar coordinates and evaluate `int_0^1 int_0^x (x+y)dydx`

Concept: Application of Double Integrals to Compute Mass

Solve by method of variation of parameters

`(d^2y)/dx^2+3 dy/dx+2y=e^(e"^x)`

Concept: Method of Variation of Parameters

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