Question Bank

Unit-1

Differential equation of first order and first degree

Question-1: Solve-

Sol. We can write the given equation as-

So that-

I.F. =

The solution of equation (1) will be-

Or

Or

Or

Question-2: Solve-

Sol.

We can write the equation as-

We see that it is a Leibnitz’s equation in x-

So that-

Therefore the solution of equation (1) will be-

Or

Question-3: : Solve

Sol.

We can write the equation as-

On dividing by , we get-

Put so that

Equation (1) becomes,

Here,

Therefore the solution is-

Or

Now put

Integrate by parts-

Or
 

Question-4: Solve

Sol. Here given,

Now let z = sec y, so that dz/dx = sec y tan y dy/dx

Then the equation becomes-

Here,

Then the solution will be-

Question-5: Solve-

Sol. Here given-

We can re-write this as-

Which is a linear differential equation-

The solution will be-

Put

Question-6: Solve-

Sol. We can write the equation as below-

Here M = and N =

So that-

The equation is exact and its solution will be-

Or

Question-7: Solve-

Sol.

We can write the given equation as-

Here,

M =

Multiply equation (1) by we get-

This is an exact differential equation-

Question-8: Solve-

Sol.

Here given,

M = 2y and N = 2x log x - xy

Then-

Here,

Then,

Now multiplying equation (1) by 1/x, we get-

Question-9: Solve-

Sol.

We can write the equation as below-

Now comparing with-

We get-

a = b = 1, m = n = 1, a’ = b’ = 2, m’ = 2, n’ = -1

I.F. =

Where-

On solving we get-

h = k = -3

Multiply the equation by , we get-

It is an exact equation.

So that the solution is-

Question-10: if the family of curves is xy = c , then find its orthogonal trajectory.

Sol.  First we will differentiate the given equation with respect to x,

       We get,

y + x   = 0

   =

Replace   by

     =

   =

We get,

Ydy = x dx

Now integrate this equation,  we get

   = + c

y² - x² = 2c.                 Ans.

Question-11: find the orthogonal trajectory of the family of curves x² - y² = c

Sol. Here we will follow same procedure as we did in above example,

      Diff. The given equation w.r.t. x, we get

                                              2x – 2y  = 0

=

Replace   by

=

= -

Ydy = - xdx

Now integrate the above eq.

   = + c

   On solving we get,

x² + y² = 2c.

Question-12: A body originally at cools down to in 20 minutes. The temperature of the air being . What will be the temperature of the body after 40 minutes from the original?

Sol.

If  is the temperature of the surroundings and that of the body at any time t, then-

On integrating-

Or

When t = 0 and

So that-

Then equation-1 becomes-

When t = 40 min, then-