2016-2017 June

Evaluate `int_0^∞ 3^(-4x^2) dx`

Concept: Legendre’S Differential Equation

Solve` (2y^2-4x+5)dx=(y-2y^2-4xy)dy`

Concept: Particular Integrals of Differential Equation

Solve the ODE `(D-1)^2 (D^2+1)^2y=0`

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

Evaluate `int_0^1( x^a-1)/log x dx`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Find the length of cycloid from one cusp to the next , where `x=a(θ + sinθ) , y=a(1-cosθ)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(D^2-3D+2) y= 2 e^x sin(x/2)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Using D.U.I.S prove that `int_0^∞ e^-(x^+a^2/x^2) dx=sqrtpi/2 e^(-2a), a> 0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Change the order of integration and evaluate `int_0^1 int_x^sqrt(2-x^2 x dx dy)/sqrt(x^2+y^2)`

Concept: Differentiation Under Integral Sign with Constant Limits of Integration

Evaluate `int_0^1int_0^( 1-x)1int_0^( 1-x-y) 1/(x+y+z+1)^3 dx dy dz`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Find the mass of the lemniscate 𝒓^{𝟐}=𝒂^{𝟐}𝒄𝒐𝒔 𝟐𝜽 if the density at any point is Proportional to the square of the distance from the pole .

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve` x^2 (d^3y)/dx^3+3x (d^2y)/dx^2+dy/dx+y/x=4log x`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Prove that for an astroid ` x^(2/3) +y2/3= a^(2/3)` the line 𝜽=𝝅/𝟔 Divide the arc in the first quadrant in a ratio 1:3.

Concept: Exact Differential Equations

Solve `(D^2-7D-6)y=(1+x^2)e^(2x)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Apply Rungee Kutta method of fourth order to find an approximate Value of y when x=0.4 given that `dy/dx=(y-x)/(y+x),y=1` 𝒚=𝟏 𝒘𝒉𝒆𝒏 𝒙=𝟎 Taking h=0.2.

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Use Taylor series method to find a solution of `dy/dx=xy+1,y(0)=0` X=0.2 taking h=0.1 correct upto 4 decimal places.

Concept: Taylor’S Series Method

Solve by variation of parameters` ((d^2y)/dx^2+1)y=1/(1+sin x)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Compute the value of `int _0.2^1.4 (sin x - In x+e^x) ` Trapezoidal Rule (ii) Simpson’s (1/3)rd rule (iii) Simpson’s (3/8)th rule by dividing Into six subintervals.

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Using beta functions evaluate `int_0^(pi/6) cos^6 3θ.sinθ dθ`

Concept: Particular Integrals of Differential Equation

Evaluate `int_0^(a/sqrt2) int_y^(sqrt(a2-y^2)) log (x^2+y^2) "dxdy by changing to polar Coordinates".`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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