NCERT Class 12-Mathematics: Chapter –9 Differential Equations Part 2

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Question 6:

Find the differential equation of all non-horizontal lines in a plane.

Answer:

The general equation of all non-horizontal lines in a plane is

.

Therefore,

Again, differentiating both sides w.r.t. y, we get

Question 7:

Find the equation of a curve whose tangent at any point on it, different from origin, has slope

Answer:

Given

Integrating both sides, we get

Long Answer (L.A.)

Question 8:

Find the equation of a curve passing through the point if the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the axis.

Answer:

Let the equation of normal at

Therefore, the length of perpendicular from origin to is

Also distance between P and axis is . Thus, we get

Case I:

Integrating both sides, we get , Substituting , we get .

Therefore, is the equation of curve (not possible, so rejected).

Case II: Substituting , we get

Integrating both sides, we get

. Substituting , , we get .

Therefore, is the required equation.

Question 9:

Find the equation of a curve passing through . if the slope of the tangent to the curve at any point is

Answer:

According to the given condition

This is a homogeneous differential equation. Substituting , we get

which is the required equation.

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