q1 find a general solution of the following

Question

Q.1  Find a general solution of the following ordinary differential equations:

 (a) y' = 3(y+1)         (b) y' = y²sinx       (c) y' = e^2xcos²y.

Q.2  Solve the following  differential equations with the given initial values.

 (a) y' = x³e^-y ,      y(2) = 0

(b)   y'cosh²x + sinh²y =0 , y(0) = -2

Plot the solutions.

3. (a)  Solve the following differential equation with the given initial value ( initial value problem:

           y' = y/x + 2x³cosx²/y    ;       y(√Π) = 0   . Plot the solution.

        b) Find the general solution of the following ordinary differential equation.

            x²y' = y²+xy

Q.4  (a) Solve the following initial value problem:

                 y' + ytanx = sin2x,  y(0) = 1

         (b) Find the general solutions of the following differential equation:

                         xy' + 2y = 9x

Q.5   Solve the following initial value problems:

                 (a)  y' + 3y = 12, y(0) = 6

Plot the solution of (a).

                 (b) y' + ky, e^-kx , y(0) = 0.7

Q.6 Find a general solution of the following system of linear differential equations:

             y₁' = y₁ + y₂    .

            y₂' = 3y₁ - y₂

Q.7 Find a general solution of the following system of linear differential equations:

            y₁' = y₁ - y₂   

             y₂' = y₁ + y₂.

Q.8 Find a general solution of the following system:

y₁" = 3y₁ -y₂ + 2y₃

y₂' = -4y₂ + 2y₃

y₃' = y₂ -5y₃