MATHEMATICS II

Correct answer is A.

Correct answer is C.

 Correct answer is B.

Correct answer is  A.

into suggested equations, we find that the only equation that is satisfied is D) indeed

Correct answer is  D .

Correct answer is E .

Integrating by parts

Therefore the general solution is

From initial condition 

hence

Correct answer is  C .

Therefore

Correct answer is  A

The characteristic equation

The general solution

Correct answer is  A.

so  the  equation

is a differential equation.

  Correct answer is  E.

Multiplying the equation

by this factor leads to the equation

By integrating both sides of above equation, using integration by parts on the second term on the right side, we obtain

where c is an arbitrary constant. Thus

Multiplying the differential equation by

we obtain

Then, by integrating both sides of this equation, we have

where we have used integration by parts on the last term in the above equation. Thus the general solution of the differential equation is

we have

so

To solve the equation

we first compute the integrating factor

On multiplying the equation

by

we obtain

and therefore

where c is an arbitrary constant. It follows that

is the general solution of differential equation.

and it then follows that m must be a root of the characteristic equation

Thus the possible values of

the general solution of the differential equation is

from the previous solution. To satisfy the first initial condition we set

 in the general solution; thus

must satisfy

 To use the second initial condition we must first  differentiate the equation

This gives

Then, setting

we obtain

By solving the equations involving

together we find that

Using these values in the general solution, we obtain the solution

then the characteristic equation is

and its roots are

Therefore the general solution of the differential equation is

Applying the initial conditions, we obtain the following two equations for

The solution of these equations is

and the solution of the initial value problem is

in the equation we get

Hence

so the general solution of the  equation is

in the Euler equation we obtain

the general solution of the given equation is

Integrating the left side with respect to

and the right side with respect to

gives

Substituting

we obtain

This equation is quadratic in y and we obtain

Only

satisfy the given initial condition.