BE Mechanical Engineering Semester 2 (FE First Year)University of Mumbai
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# Applied Mathematics 2 CBCGS 2018-2019 BE Mechanical Engineering Semester 2 (FE First Year) Question Paper Solution

SubjectApplied Mathematics 2
Year2018 - 2019 (December)
Applied Mathematics 2
CBCGS
2018-2019 December
Marks: 80

Question no 1 compulsory

Attempt  ant tree Questions from remaining five questions.

[20]1
[3]1.a

Evaluate int_0^inftye^(x^3)/sqrtx 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.b

Find the length of the curve x=y^3/3+1/(4y) from y=1 to y=2

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.c

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

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

Solve (4x+3y-4)dx+(3x-7y-3)dy=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.f

Solve dy/dx=1+xy with initial condition x_0=0,y_0=0.2 By Taylors series method. Find the approximate value of y for x= 0.4(step size = 0.4).

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^2y)/dx^2-16y=x^2 e^(3x)+e^(2x)-cos3x+2^x

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

Show that int_0^pi log(1+acos x)/cos x dx=pi sin^-1 a  0 ≤ a ≤1.

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_0^2 int_(2-sqrt(4-y^2))^(2+sqrt(4-y^2)) dxdy

Chapter: [10] Triple Integration and Applications of Multiple Integrals
Concept: Application of Double Integrals to Compute Volume
[20]3
[6]3.a

Evaluate int int int (x+y+z) dxdydz  over the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1.

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 lamina bounded by the curves 𝒚 = 𝒙𝟐 − 𝟑𝒙 and 𝒚 = 𝟐𝒙 if the density of the lamina at any point is given by 24/25 xy

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.c

Solve x^2 (d^2y)/dx^2+3x dy/dx+3y =(log x.cos (log x))/x

Chapter: [5] Differential Equations of First Order and First Degree
Concept: Equations Reducible to Exact Form by Using Integrating Factors
[20]4
[8]4.a

Find by double integration the area bounded by the parabola 𝒚𝟐=𝟒𝒙 And 𝒚=𝟐𝒙−𝟒

Chapter: [10] Triple Integration and Applications of Multiple Integrals
Concept: Application of Double Integrals to Compute Area
[6]4.b

Solve dy/dx+x sin 2 y=x^3 cos^2 y

Chapter: [7] Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Concept: Runga‐Kutta Fourth Order Formula
[6]4.c

Solve dy/dx=x^3+ywith initial conditions y(0)=2 at x= 0.2 in step of h = 0.1 by Runge Kutta method of Fourth order.

Chapter: [7] Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Concept: Runga‐Kutta Fourth Order Formula
[20]5
[6]5.a

Evaluate int_0^1 x^5 sin ^-1 x dxand find the value of β (9/2,1/2)

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

In a circuit containing inductance L, resistance R, and voltage E, the current i is given by L (di)/dt+Ri=E.Find the current i at time t at t = 0 and i = 0 and L, R and E are constants.

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

Evaluate int_0^6 dx/(1+3x)by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule.

Chapter: [5] Differential Equations of First Order and First Degree
Concept: Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem
[20]6
[6]6.a

Find the volume bounded by the paraboloid 𝒙𝟐+𝒚𝟐=𝒂𝒛 and the cylinder 𝒙𝟐+𝒚𝟐=𝒂𝟐.

Chapter: [10] Triple Integration and Applications of Multiple Integrals
Concept: Application of Double Integrals to Compute Mass
[6]6.b

Change to polar coordinates and evaluate int_0^1 int_0^x (x+y)dydx

Chapter: [10] Triple Integration and Applications of Multiple Integrals
Concept: Application of Double Integrals to Compute Mass
[8]6.c

Solve by method of variation of parameters

(d^2y)/dx^2+3 dy/dx+2y=e^(e"^x)

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

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