# Applied Mathematics 2 CBCGS 2017-2018 BE Production Engineering Semester 2 (FE First Year) Question Paper Solution

Applied Mathematics 2 [CBCGS]
Date: December 2017

(1) Question no. 1 is compulsory.
(2) Attempt any 3 questions from remaining five questions.

 1
 1.a

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

Concept: Beta and Gamma Functions and Its Properties
Chapter:  Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
 1.b

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

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

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

Concept: Exact Differential Equations
Chapter:  Differential Equations of First Order and First Degree
 1.d

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
Chapter:  Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
 1.e

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
Chapter:  Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
 1.f

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

Concept: Rectification of Plane Curves
Chapter:  Differentiation Under Integral Sign, Numerical Integration and Rectification
 2
 2.a

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

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

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
Chapter:  Double Integration
 2.c

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
Chapter:  Triple Integration and Applications of Multiple Integrals
 3
 3.a

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
Chapter:  Triple Integration and Applications of Multiple Integrals
 3.b

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
Chapter:  Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
 3.c

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
Chapter:  Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
 4
 4.a

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
Chapter:  Differentiation Under Integral Sign, Numerical Integration and Rectification
 4.b

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

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

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
Chapter:  Triple Integration and Applications of Multiple Integrals
 5
 5.a

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

Concept: Equations Reducible to Exact Form by Using Integrating Factors
Chapter:  Differential Equations of First Order and First Degree
 5.b

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
Chapter:  Triple Integration and Applications of Multiple Integrals
 5.c

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

Concept: Numerical Integration‐ by Trapezoidal
Chapter:  Differentiation Under Integral Sign, Numerical Integration and Rectification

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
Chapter:  Differentiation Under Integral Sign, Numerical Integration and Rectification

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
Chapter:  Differentiation Under Integral Sign, Numerical Integration and Rectification
 6
 6.a

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
Chapter:  Double Integration
 6.b

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
Chapter:  Triple Integration and Applications of Multiple Integrals
 6.c

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

Concept: Method of Variation of Parameters
Chapter:  Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order

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