NCERT Solutions for Maths Class 12 Exercise 9.3

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Class 12th is a very crucial stage of your student’s life, since you take all important decisions about your career on this stage. Mathematics plays a vital role to take decision for your career because if you are good in mathematics, you can choose engineering and technology field as your career.

NCERT Solutions for Maths Class 12 Exercise 9.3 helps you to solve each and every problem with step by step explanation which makes you strong in mathematics.

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NCERT Solutions for Maths Class 12 Exercise 9.3 are prepared by the experienced teachers of CBSE board. If you are preparing for JEE Mains and NEET level exams, then it will definitely make your foundation strong.

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NCERT Solutions for Maths Class 12 Exercise 9.1

NCERT Solutions for Maths Class 12 Exercise 9.2

NCERT Solutions for Maths Class 12 Exercise 9.4

NCERT Solutions for Maths Class 12 Exercise 9.5

NCERT Solutions for Maths Class 12 Exercise 9.6

NCERT Solutions for Maths Class 12 Exercise 9.3

In each of the Exercises 1 to 5, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

Maths Class 12 Ex 9.3 Question 1.

Solution:

The given equation of the curve is …(i)

Differentiating equation (i) w.r.t. x, we get
1/a + y’/b = 0 …(ii)
Again differentiating equation (ii) w.r.t. x, we get
y’’/b = 0 ⇒ y’’ = 0
Hence, the differential equation representing the given family of curve is y’’ = 0.

Maths Class 12 Ex 9.3 Question 2.

y² = a (b² – x²)

Solution:

The given equation of the curve is y² = a (b² – x²)           … (i)
Differentiating equation (i) w.r.t. x, we get
2yy’ = a(–2x)      ⇒      yy’ = –ax        …(ii)
Again differentiating equation (ii) w.r.t. x, we get
yy’’ + y’. y’ = –a     ⇒    yy’’ + y’² = –a       … (iii)

Eliminating a from equation (ii) and (iii), we get

yy’ = (yy’’ + y’²)x ⇒ xy y’’ + xy’² – yy’ = 0
Hence, the differential equation representing the given family of curve is xy y’’ + xy’² – yy’ = 0.

Maths Class 12 Ex 9.3 Question 3.

y = a e^3x + b e^-2x

Solution:

The given equation of the curve is y = a e^3x + b e^-2x … (i)
Differentiating equation (i) w.r.t. x, we get

y’ = a e^3x × 3 + b e^-2x× (–2) … (ii)

Multiplying equation (i) by 3 and subtracting it from equation (ii), we get

y’ – 3y = –5b e^-2x … (iii)

Differentiating equation (iii) w.r.t. x, we get

y’’ – 3y’ = –5b e^-2x× (–2) … (iv)

Multiplying equation (iii) by 2 and adding it with equation (iv), we get

y’’ – 3y’ + 2(y’ – 3y) = 0 ⇒ y’’ – y’ – 6y = 0

Hence, the differential equation representing the given family of curve is y’’ – y’ – 6y = 0.

Maths Class 12 Ex 9.3 Question 4.

y = e^2x (a + bx)

Solution:

The given equation of the curve is y = e^2x (a + bx) … (i)

Differentiating equation (i) w.r.t. x, we get

y’ = a e^2x × 2 + b e^2x+ bx e^2x × 2

y’ = b e^2x+ 2 e^2x (a + bx)

y’ = b e^2x+ 2y ⇒ y’ – 2y = b e^2x … (ii)

Differentiating equation (ii) w.r.t. x, we get

y’’ – 2y’ = 2b e^2x … (iii)

From equations (ii) and (iii), we get

y’’ – 2y’ = 2(y’ – 2y) ⇒ y’’ – 4y’ + 4y = 0

Hence, the differential equation representing the given family of curve is y’’ – 4y’ + 4y = 0.

Maths Class 12 Ex 9.3 Question 5.

y = e^x (a cos x + b sin x)

Solution:

The given equation of the curve is y = e^x (a cos x + b sin x) … (i)

Differentiating equation (i) w.r.t. x, we get

Maths Class 12 Ex 9.3 Question 6.

Form the differential equation of the family of circles touching the y-axis at origin.

Solution:

The equation of the circle with centre (a, 0) and radius a, which touches y-axis at origin is

Maths Class 12 Ex 9.3 Question 7.

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Solution:

The equation of parabola having vertex at the origin and axis along positive y-axis is

Maths Class 12 Ex 9.3 Question 8.

Form the differential equation of family of ellipses having foci on y-axis and centre at origin.

Solution:

The equation of family of ellipses having foci at y- axis and centre at origin is

Maths Class 12 Ex 9.3 Question 9.

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.

Solution:

The equation of the family of hyperbola is
Differentiating both sides w.r.t, x, we get

which is the required differential equation of the hyperbola.

Maths Class 12 Ex 9.3 Question 10.

Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

Solution:

Let centre of the circle be (0, a) and radius r = 3 units
Equation of the circle is x² + (y – a)² = 9 … (i)
Differentiating equation (i) w.r.t. x, we get