Find the solution of the exact differential equation and

Question

Question 1: The bending moment, M, of a beam of length l is given by:

dM/dx = w(x + l) ,

where w is the load constant. If M(0) = 0, find M as a function of x.

Question 2: Solve the following separable differential equations:

(i) y' = (1+x)(1+y), (y > -1),

(ii) y' = 2y^1/2 - 2y / x and (iii) y' = ycosx /1 + y²

Question 3: Solve the following initial value problem:

y' = x² √(9 - x²) e^ycosy, y(0) = 0.

Question 4: Which of the following differential equations is exact:

(a) 2xy³dx+ (1 + 3x²y²)dy = 0  (b) x²y dx + 5xy²dy = 0

Find the solution of the exact differential equation.

Question 5: Solve the following differential equation with the given initial value.

2x sin ydx+ (x²cosy - 1) dy = 0, y(0) = 0.

Question 6: The differential equation : P(x,y) dx + Q (x,y) dy = 0 is not exact. However, if it is multiplied by an integrating factor F(x), it can be made exact. Show that the integrating factor F(x) is given by the solution of the following differential equation:

1/F dF/dx = 1/Q (∂p/∂y - ∂Q/∂x )

Question 7: First find the integrating factor of the following differential equation and then find its general solution:

2coshx cosydx = sinhx siny dy.

Question 8: The following ODEs are in the form of yl + p(x)y = r(x) , find their solutions.

(i) y' +3y = e^x ,y(0) =0 (ii) (1-x²))y' -2xy =1 (|x| ≠ 1), y(0) = 0.

Question 9: Find a homogeneous second order linear ODE of the form of :

y''+ ay' + by = 0 which has the basis y₁ = e^2x , y₂ = e^x.