Module-4

Ordinary differential equations of first order

Question-1: Solve

Sol.

Here M = and N =

Then the equation is exact and its solution is-

Question-2: Determine whether the differential function ydx –xdy = 0 is exact or not.

Solution.  Here the equation is the form of  M(x , y)dx + N(x , y)dy = 0

  But, we will check for exactness,

These are not equal results, so we can say that the given diff. Eq. Is not exact.

Question-3: Solve-

Sol.

We can write the given equation as-

Here,

M =

Multiply equation (1) by we get-

This is an exact differential equation-

Question-4: 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-5: 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-6: Solve-

Sol. Here given-

We can re-write this as-

Which is a linear differential equation-

The solution will be-

Put

Question-7: 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-8: If a body which is at the temperature of cools down to within 20 minutes. The temperature of the air is .

Find that what will be the temperature of the body after 40 min from its original temperature?

Sol.

Suppose  be the temperature of the body at time ‘t’ then-

Here k is the constant.

On integrating, we get-

‘c’ is the constant

Or

Or

When t = 0, and when t = 20 then

So that- and

Then equation (1) becomes-

Now, when t = 40 minutes, then-

Question-9: 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-10: Solve-

Sol.

Here we have-

Divide this by (x + 1), we get-

Which is the Leibnitz’s equation-

Here-

And

Integrating factor-

The solution will be-

Or

Question-11: Solve

Sol.

Here we have-

Now differentiate it with respect to x, we get-

Or

This is the Leibnitz’s linear equation in x and p, here

Then the solution of (2) is-

Or

Or

Put this value of x in (1), we get

Question-12: Solve-

Sol.

Put

So that-

Then the given equation becomes-

Or

Or

Which is the Clairaut’s form.

Its solution is-

i.e.

Module-4

Ordinary differential equations of first order

Question-1: Solve

Sol.

Here M = and N =

Then the equation is exact and its solution is-

Question-2: Determine whether the differential function ydx –xdy = 0 is exact or not.

Solution.  Here the equation is the form of  M(x , y)dx + N(x , y)dy = 0

  But, we will check for exactness,

These are not equal results, so we can say that the given diff. Eq. Is not exact.

Question-3: Solve-

Sol.

We can write the given equation as-

Here,

M =

Multiply equation (1) by we get-

This is an exact differential equation-

Question-4: 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-5: 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-6: Solve-

Sol. Here given-

We can re-write this as-

Which is a linear differential equation-

The solution will be-

Put

Question-7: 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-8: If a body which is at the temperature of cools down to within 20 minutes. The temperature of the air is .

Find that what will be the temperature of the body after 40 min from its original temperature?

Sol.

Suppose  be the temperature of the body at time ‘t’ then-

Here k is the constant.

On integrating, we get-

‘c’ is the constant

Or

Or

When t = 0, and when t = 20 then

So that- and

Then equation (1) becomes-

Now, when t = 40 minutes, then-

Question-9: 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-10: Solve-

Sol.

Here we have-

Divide this by (x + 1), we get-

Which is the Leibnitz’s equation-

Here-

And

Integrating factor-

The solution will be-

Or

Question-11: Solve

Sol.

Here we have-

Now differentiate it with respect to x, we get-

Or

This is the Leibnitz’s linear equation in x and p, here

Then the solution of (2) is-

Or

Or

Put this value of x in (1), we get

Question-12: Solve-

Sol.

Put

So that-

Then the given equation becomes-

Or

Or

Which is the Clairaut’s form.

Its solution is-

i.e.