Differential Equations - 18 - Mathematics MCQ

The given differential equations are
  ......(i)

The infinite number of differential equations given by

for infinitely many different choices of Q(x), possess the same integrating factor 

The given ciifferential equation is

The required Integrating factor is given by
   [log x = t]
= e ^{log t}  = t = log x

The given differential equation is

The required Integrating factor is given by

The given differential equation can be rewritten as

This implies that

∴ differential equation (i) reduces to

Remark : Observe that the transformation (ii) is suggested by form (i) of given differential eouation.

Rewrite the given differential equation as

This suggests that the required transformation is

The given differential equation is
 .....(i)
Its Integrating Factor is given by


becomes exact.
This implies

The differential equation is x dy = y dx

This represents the family of straight lines passing through the origin.

Statement (c) gives the correct definition of an isogonal trajectory.

Given family of the curves is
   .....(i)
Its slope is dy/dx. If m is the slope of the orihogonal trajectories of the family (i) then we should have

Then given family of curves is
y = cx^k  ....(i)
.v —
Let us first find the differential equation satisfied by the family (i). For this we differentiate (i) w.r. t. 

∴ The differential equation of the orthogonal trajectories will be obtained on replacing 
⇒ Orthogonal trajectories are given by

 The given differential equation 

Hint: Write

and do as in the previous question.

Some comments before the solution.
1. If the given equation in x, y and p can be solved for y, then we shall have
y = f(x,p)    ...(i)
Differentiate (i) w.r. t. x. This will lead to a differentia! equation in p and x. Solve it for p and then eliminate p from the resulting equation and (i).
2. If the given equation in x, y and p can be solved for x, then we shail have
x = f(y, p) .......(ii)
Differentiate (ii), w.r. t. y and follow the same procedure as in case - i.
Now solution of the problem 82.
The given differential equation is
y = 2 px + p²y   ...... (iii)
  ... (iv)
which is solved for x, i.e. it expresses x as a function of y and p.
Now differentiating (iv) w.r. t. y, we get


Now eliminating p from equations (iii) and (v), we get

or y² = 2cx + c²

The given differential equation is




Now eliminating p from equations (i) and (iii).