# Question Bank Solutions for BE Biotechnology Semester 2 (FE First Year) - University of Mumbai - Applied Mathematics 2

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Applied Mathematics 2
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Evaluate int_0^oo5^(-4x^2)dx

 Differential Equations of First Order and First Degree
Chapter:  Differential Equations of First Order and First Degree
Concept: Exact Differential Equations

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

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

Evaluate int_0^1sqrt(sqrtx-x)dx

 Differentiation Under Integral Sign, Numerical Integration and Rectification
Chapter:  Differentiation Under Integral Sign, Numerical Integration and Rectification
Concept: Differentiation Under Integral Sign with Constant Limits of Integration

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

 Differential Equations of First Order and First Degree
Chapter:  Differential Equations of First Order and First Degree
Concept: Exact Differential Equations

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

 Double Integration
Chapter:  Double Integration
Concept: Double Integration‐Definition

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

 Triple Integration and Applications of Multiple Integrals
Chapter:  Triple Integration and Applications of Multiple Integrals
Concept: Triple Integration Definition and Evaluation

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).

 Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Chapter:  Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Concept: Taylor’S Series Method

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.

 Triple Integration and Applications of Multiple Integrals
Chapter:  Triple Integration and Applications of Multiple Integrals
Concept: Triple Integration Definition and Evaluation

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

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

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.

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

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

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

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

 Differential Equations of First Order and First Degree
Chapter:  Differential Equations of First Order and First Degree
Concept: Exact Differential Equations

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.

 Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Chapter:  Numerical Solution of Ordinary Differential Equations of First Order and First Degree, Beta and Gamma Function
Concept: Taylor’S Series Method

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

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

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

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

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

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

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.

 Triple Integration and Applications of Multiple Integrals
Chapter:  Triple Integration and Applications of Multiple Integrals
Concept: Triple Integration Definition and Evaluation

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

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

Solve (1+x)^2(d^2y)/(dx^2)+(1+x)(dy)/(dx)+y=4cos(log(1+x))

 Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
Chapter:  Linear Differential Equations with Constant Coefficients and Variable Coefficients of Higher Order
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)

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