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

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Applied Mathematics 2
CBCGS
2017-2018 June
Marks: 80

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


[20]1
[3]1.a

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

Concept: Exact Differential Equations
Chapter: [5] Differential Equations of First Order and First Degree
[3]1.b

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
Chapter: [7] Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
[3]1.c

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

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

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

Concept: Differentiation Under Integral Sign with Constant Limits of Integration
Chapter: [8] Differentiation Under Integral Sign, Numerical Integration and Rectification
[4]1.e

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

Concept: Exact Differential Equations
Chapter: [5] Differential Equations of First Order and First Degree
[4]1.f

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

Concept: Double Integration‐Definition
Chapter: [9] Double Integration
[20]2
[6]2.a

Solve  `xy(1+xy^2)(dy)/(dx)=1`

Concept: Linear Differential Equations
Chapter: [5] Differential Equations of First Order and First Degree
[6]2.b

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
Chapter: [10] Triple Integration and Applications of Multiple Integrals
[8]2.c

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
Chapter: [7] Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
[20]3
[6]3.a

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

Concept: Rectification of Plane Curves
Chapter: [8] Differentiation Under Integral Sign, Numerical Integration and Rectification
[6]3.b

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
Chapter: [9] Double Integration
[8]3.c

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

Concept: Triple Integration Definition and Evaluation
Chapter: [10] Triple Integration and Applications of Multiple Integrals
[20]4
[6]4.a

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

Concept: Method of Variation of Parameters
Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
[6]4.b

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
Chapter: [7] Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
[8]4.c

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
Chapter: [8] Differentiation Under Integral Sign, Numerical Integration and Rectification

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

Concept: Numerical Integration‐ by Trapezoidal
Chapter: [8] Differentiation Under Integral Sign, Numerical Integration and Rectification

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
Chapter: [8] Differentiation Under Integral Sign, Numerical Integration and Rectification
[20]5
[6]5.a

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

Concept: Equations Reducible to Exact Form by Using Integrating Factors
Chapter: [5] Differential Equations of First Order and First Degree
[6]5.b

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
Chapter: [10] Triple Integration and Applications of Multiple Integrals
[8]5.c

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

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

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

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

Concept: Method of Variation of Parameters
Chapter: [6] Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
[8]6.c

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
Chapter: [10] Triple Integration and Applications of Multiple Integrals

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