BE Marine Engineering छमाही २ (इंजीनियरिंग) - University of Mumbai Important Questions for Applied Mathematics 2

Solve` (2y^2-4x+5)dx=(y-2y^2-4xy)dy` 

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

Solve the ODE `(D-1)^2 (D^2+1)^2y=0` 

Evaluate `int_0^1( x^a-1)/log x dx` 

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

Find the length of cycloid from one cusp to the next , where `x=a(θ + sinθ) , y=a(1-cosθ)`

Solve `(D^2-3D+2) y= 2 e^x sin(x/2)`

Using D.U.I.S prove that `int_0^∞ e^-(x^+a^2/x^2) dx=sqrtpi/2 e^(-2a), a> 0` 

Evaluate `int_0^1int_0^( 1-x)1int_0^( 1-x-y)     1/(x+y+z+1)^3 dx dy dz` 

Find the mass of the lemniscate 𝒓^𝟐=𝒂^𝟐𝒄𝒐𝒔 𝟐𝜽 if the density at any point is Proportional to the square of the distance from the pole . 

Solve`  x^2 (d^3y)/dx^3+3x (d^2y)/dx^2+dy/dx+y/x=4log x` 

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

Solve `(D^2-7D-6)y=(1+x^2)e^(2x)`

Apply Rungee Kutta method of fourth order to find an approximate Value of y when x=0.4 given that `dy/dx=(y-x)/(y+x),y=1` 𝒚=𝟏 𝒘𝒉𝒆𝒏 𝒙=𝟎 Taking h=0.2. 

Solve by variation of parameters` ((d^2y)/dx^2+1)y=1/(1+sin x)`

Compute the value of `int _0.2^1.4 (sin  x - In x+e^x) ` Trapezoidal Rule (ii) Simpson’s (1/3)rd rule (iii) Simpson’s (3/8)th rule by dividing Into six subintervals. 

Using beta functions evaluate `int_0^(pi/6)  cos^6 3θ.sinθ dθ` 

Evaluate `int_0^(a/sqrt2) int_y^(sqrt(a2-y^2)) log (x^2+y^2) "dxdy by changing to polar Coordinates".` 

Evaluate `int int int  x^2` `yzdzdydz`over the volume bounded by planes x=0, y=0, z=0 and `x/a+y/b+z/c=1`

Evaluate `int_0^inftye^(x^3)/sqrtx dx`