Chapter 9 Differential Equations Ex 9.3 Solutions
Question - 1 : - 
Answer - 1 : -
Differentiating both sides of the given equation with respect to x, we get:
Again, differentiating both sides with respect to x, we get:
Hence, the required differential equation of the given curve is 
Answer - 2 : -
Differentiating both sides of the given equation with respect to x, we get:
Again, differentiating both sides with respect to x, we get:
Hence, the required differential equation of the given curve is 
Answer - 3 : -
Differentiating both sides with respect to x, we get:
Again, differentiating both sides with respect to x, we get:
Multiplying equation (1) with (2) and then adding it to equation (2), we get:
Now, multiplying equation (1) with 3 and subtracting equation (2) from it, we get:
Substituting the values of 
in equation (3), we get: This is the required differential equation of the given curve.
Question - 4 : - 
Answer - 4 : -
From the question it is given that y = e2x (a+ b x) … [we call it as equation (i)]
Differentiating both sides with respect to x, we get,
y’ = 2e2x(a + b x) + e2x × b … [equation (ii)]
Then, multiply equation (i) by 2 and afterwards subtract it toequation (ii),
We have,
y’ – 2y = e2x(2a + 2bx+ b) – e2x (2a + 2bx)
y’ – 2y = 2ae2x +2e2xbx + e2xb– 2ae2x – 2bxe2x
y’ – 2y = be2x …[equation (iii)]
Now, differentiating equation (iii) both sides,
We have,
⇒ y’’ –2y = 2be2x … [equation (iv)]
Then,
Question - 5 : - 
Answer - 5 : -
Differentiating both sides with respect to x, we get:
Again, differentiating with respect to x, we get:
Adding equations (1) and (3), we get:
This is the required differential equation of the given curve.
Answer - 6 : -
The centre of the circle touching the y-axis at origin lies on the x-axis.
Let (a, 0) be the centre of the circle.
Since it touches the y-axis at origin, its radius is a.
Now, the equation of the circle with centre (a, 0) and radius (a) is
Differentiating equation (1) with respect to x, we get:
Now, on substituting the value of a in equation (1), we get:
This is the required differential equation.
Answer - 7 : - The equation of the parabola having the vertex at origin and the axis along the positive y-axis is:
Differentiating equation (1) with respect to x, we get:
Dividing equation (2) by equation (1), we get:
This is the required differential equation.
Answer - 8 : - The equation of the family of ellipses having foci on the y-axis and the centre at origin is as follows:
Differentiating equation (1) with respect to x, we get:
Again, differentiating with respect to x, we get:
Substituting this value in equation (2), we get:
This is the required differential equation.
Answer - 9 : - The equation of the family of hyperbolas with the centre at origin and foci along the x-axis is:
Differentiating both sides of equation (1) with respect to x, we get:
Again, differentiating both sides with respect to x, we get:
Substituting the value of 
in equation (2), we get: This is the required differential equation.
Answer - 10 : -
Let the centre of the circle on y-axis be (0, b).
The differential equation of the family of circles with centre at (0, b) and radius 3 is as follows:
Differentiating equation (1) with respect to x, we get:
Substituting the value of (y – b) in equation (1), we get:
This is the required differential equation.