2017-2018 December

(1) Question no. 1 is compulsory.

(2) Attempt any 3 questions from remaining five questions.

Evaluate `int_0^oo e^(-x^2)/sqrtxdx`

Concept: Beta and Gamma Functions and Its Properties

Solve `(D^3+1)^2y=0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve the ODE `(y+1/3y^3+1/2x^2)dx+(x+xy^2)dy=0`

Concept: Exact Differential Equations

Use Taylor’s series method to find a solution of `(dy)/(dx) =1+y^2, y(0)=0` At x = 0.1 taking h=0.1 correct upto 3 decimal places.

Concept: Taylor’S Series Method

Given `int_0^x 1/(x^2+a^2) dx=1/atan^(-1)(x/a)`using DUIS find the value of `int_0^x 1/(x^2+a^2) `

Concept: Method of Variation of Parameters

Find the perimeter of the curve r=a(1-cos 𝜽)

Concept: Rectification of Plane Curves

Solve `(D^3+D^2+D+1)y=sin^2x`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Change the order of integration `int_0^aint_sqrt(a^2-x^2)^(x+3a)f(x,y)dxdy`

Concept: Change the Order of Integration

Evaluate `int int(2xy^5)/sqrt(x^2y^2-y^4+1)dxdy`, where R is triangle whose vertices are (0,0),(1,1),(0,1).

Concept: Application of Double Integrals to Compute Area

Find the volume enclosed by the cylinder `y^2=x` and `y=x^2` Cut off by the planes z = 0, x+y+z=2.

Concept: Triple Integration Definition and Evaluation

Using Modified Eulers method ,find an approximate value of y At x = 0.2 in two step taking h=0.1 and using three iteration Given that `(dy)/(dx)=x+3y` , y = 1 when x = 0.

Concept: Modified Euler Method

Solve `(1+x)^2(d^2y)/(dx^2)+(1+x)(dy)/(dx)+y=4cos(log(1+x))`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Show that `int_0^asqrt(x^3/(a^3-x^3))dx=a(sqrtxgamma(5/6))/(gamma(1/3))`

Concept: Differentiation Under Integral Sign with Constant Limits of Integration

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

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Use polar co ordinates to evaluate `int int (x^2+y^2)^2/(x^2y^2)` 𝒅𝒙 𝒅𝒚 over yhe area Common to circle `x^2+y^2=ax "and" x^2+y^2=by, a>b>0`

Concept: Application of Double Integrals to Compute Area

Solve `ydx+x(1-3x^2y^2)dy=0`

Concept: Equations Reducible to Exact Form by Using Integrating Factors

Find the mass of a lamina in the form of an ellipse `x^2/a^2+y^2/b^2=1`, If the density at any point varies as the product of the distance from the

The axes of the ellipse.

Concept: Application of Double Integrals to Compute Mass

Compute the value of `int_0^(pi/2) sqrt(sinx+cosx) dx` usingTrapezoidal rule by dividing into six Subintervals.

Concept: Numerical Integration‐ by Trapezoidal

Compute the value of `int_0^(pi/2) sqrt(sinx+cosx) dx` using Simpson’s (1/3)^{rd} rule by dividing into six Subintervals.

Concept: Numerical Integration‐ by Simpson’S 1/3rd

Compute the value of `int_0^(pi/2) sqrt(sinx+cosx) dx` using Simpson’s (3/8)^{th} rule by dividing into six Subintervals.

Concept: Numerical Integration‐ by Simpson’S 3/8th Rule

Change the order of Integration and evaluate `int_0^2int_sqrt(2y)^2 x^2/(sqrtx^4-4y^2)dxdy`

Concept: Change the Order of Integration

Evaluate `int int intx^2dxdydz` over the volume bounded by planes x=0,y=0, z=0 and `x/a+y/b+z/c=1`

Concept: Application of Triple Integral to Compute Volume

Solve by method of variation of parameters :`(D^2-6D+9)y=e^(3x)/x^2`

Concept: Method of Variation of Parameters

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