BE Mechanical Engineering Semester 2 (FE First Year)University of Mumbai
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Applied Mathematics 2 CBCGS 2016-2017 BE Mechanical Engineering Semester 2 (FE First Year) Question Paper Solution

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Applied Mathematics 2
CBCGS
2016-2017 June
Marks: 80

[20]1
[3]1.a

Evaluate `int_0^∞ 3^(-4x^2) dx` 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Legendre’S Differential Equation
[3]1.b

Solve` (2y^2-4x+5)dx=(y-2y^2-4xy)dy` 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Particular Integrals of Differential Equation
[3]1.c

Solve the ODE `(D-1)^2 (D^2+1)^2y=0` 

 

 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[3]1.d

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[3]1.e

Evaluate `int_0^1( x^a-1)/log x dx` 

 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[3]1.f

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[20]2
[6]2.a

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[6]2.b

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[8]2.c

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

Chapter: [8] Differentiation Under Integral Sign, Numerical Integration and Rectification
Concept: Differentiation Under Integral Sign with Constant Limits of Integration
[20]3
[6]3.a

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

 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[6]3.b

Find the mass of the lemniscate 𝒓𝟐=𝒂𝟐𝒄𝒐𝒔 𝟐𝜽 if the density at any point is Proportional to the square of the distance from the pole . 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[8]3.c

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

 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[20]4
[6]4.a

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. 

 

Chapter: [5] Differential Equations of First Order and First Degree
Concept: Exact Differential Equations
[6]4.b

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[8]4.c

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. 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[20]5
[6]5.a

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. 

Chapter: [7] Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Concept: Taylor’S Series Method
[6]5.b

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[8]5.c

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. 

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[20]6
[6]6.a

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Particular Integrals of Differential Equation
[6]6.b

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

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
[8]6.c

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`

Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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