BE Mechanical Engineering सत्र २ (इंजीनियरिंग) - University of Mumbai Important Questions

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).

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

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

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.

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` 

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.

Solve by method of variation of parameters 

`(d^2y)/dx^2+3 dy/dx+2y=e^(e"^x)` 

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)

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.

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).

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

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.

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.

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. 

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

Solve `dy/dx+x sin 2 y=x^3 cos^2 y` 

Expand 2 𝒙³ + 7 𝒙² + 𝒙 – 6 in power of (𝒙 – 2) by using Taylors Theorem.

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

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

Find the value of the integral `int_0^1 x^2/(1+x^3`𝒅𝒙 using Simpson’s (𝟑/𝟖)^𝒕𝒉 rule.