2017-2018 June

(1) Question no. 1 is compulsory.

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

Evaluate `int_0^oo5^(-4x^2)dx`

Concept: Exact Differential Equations

Solve `dy/dx=x.y` with help of Euler’s method ,given that y(0)=1 and find y when x=0.3

(Take h=0.1)

Concept: Euler’S Method

Evaluate `(d^4y)/(dx^4)+2(d^2y)/(dx^2)+y=0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Evaluate `int_0^1sqrt(sqrtx-x)dx`

Concept: Differentiation Under Integral Sign with Constant Limits of Integration

Solve : `(1+log x.y)dx +(1+x/y)`dy=0

Concept: Exact Differential Equations

Evaluate I = `int_0^1 int_0^(sqrt(1+x^2)) (dx.dy)/(1+x^2+y^2)`

Concept: Double Integration‐Definition

Solve `xy(1+xy^2)(dy)/(dx)=1`

Concept: Linear Differential Equations

Find the area inside the circle r=a sin𝜽 and outside the cardioide r=a(1+cos𝜽 )

Concept: Application of Double Integrals to Compute Area

Apply Rungee-Kutta Method of fourth order to find an approximate value of y when x=0.2 given that `(dy)/(dx)=x+y` when y=1 at x=0 with step size h=0.2.

Concept: Runga‐Kutta Fourth Order Formula

Show that the length of curve `9ay^2=x(x-3a)^2 "is" 4sqrt3a`

Concept: Rectification of Plane Curves

Change the order of integration of `int_0^1int_(-sqrt(2y-y^2))^(1+sqrt(1-y^2)) f(x,y)dxdy`

Concept: Change the Order of Integration

Find the volume of the paraboloid `x^2+y^2=4z` cut off by the plane 𝒛=𝟒

Concept: Triple Integration Definition and Evaluation

Show that `int_0^1(x^a-1)/logx dx=log(a+1)`

Concept: Method of Variation of Parameters

If 𝒚 satisfies the equation `(dy)/(dx)=x^2y-1` with `x_0=0, y_0=1` using Taylor’s Series Method find 𝒚 𝒂𝒕 𝒙= 𝟎.𝟏 (take h=0.1).

Concept: Taylor’S Series Method

Find the value of the integral `int_0^1 x^2/(1+x^3`𝒅𝒙 using Simpson’s (𝟑/𝟖)^{𝒕𝒉} rule.

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

Find the value of the integral `int_0^1 x^2/(1+x^3`𝒅𝒙 using Trapezoidal rule

Concept: Numerical Integration‐ by Trapezoidal

Find the value of the integral `int_0^1 x^2/(1+x^3`𝒅𝒙 using Simpson’s (1/3)^{𝒕𝒉} rule.

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

Solve `(y-xy^2)dx-(x+x^2y)dy=0`

Concept: Equations Reducible to Exact Form by Using Integrating Factors

Evaluate `int int int sqrt(1-x^2/a^2-y^2/b^2-x^2/c^2 )`dx dy dz over the ellipsoid `x^2/a^2+y^2/b^2+z^2/c^2=1.`

Concept: Triple Integration Definition and Evaluation

Evaluate `(2x+1)^2(d^2y)/(dx^2)-2(2x+1)(dy)/(dx)-12y=6x`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

A resistance of 100 ohms and inductance of 0.5 henries are connected in series With a battery of 20 volts. Find the current at any instant if the relation between L,R,E is L `(di)/(dt)+Ri=E.`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve by variation of parameter method `(d^2y)/(dx^2)+3(dy)/(dx)+2y=e^(e^x)`.

Concept: Method of Variation of Parameters

Evaluate `int int xy(x-1)dx dy` over the region bounded by 𝒙𝒚 = 𝟒,𝒚= 𝟎,𝒙 =𝟏 and 𝒙 = 𝟒

Concept: Application of Double Integrals to Compute Area

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