NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3.

- Differential Equations Class 12 Ex 9.1
- Differential Equations Class 12 Ex 9.2
- Differential Equations Class 12 Ex 9.4
- Differential Equations Class 12 Ex 9.5
- Differential Equations Class 12 Ex 9.6

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 12 |

Subject |
Maths |

Chapter |
Chapter 9 |

Chapter Name |
Differential Equations |

Exercise |
Ex 9.3 |

Number of Questions Solved |
12 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3

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

**Ex 9.3 Class 12 Maths Question 1.**

\(\frac { x }{ a } +\frac { y }{ b } =1\)

**Solution:**

Given that \(\frac { x }{ a } +\frac { y }{ b } =1\) …(i)

differentiating (i) w.r.t x, we get

\(\frac { 1 }{ a } +\frac { 1 }{ b } { y }^{ I }=0\) …(ii)

again differentiating w.r.t x, we get

\(\frac { 1 }{ b } { y }^{ II }=0\Rightarrow { y }^{ II }=0\)

which is the required differential equation

**Ex 9.3 Class 12 Maths Question 2.**

y² = a(b² – x²)

**Solution:**

given that

y² = a(b² – x²)…(i)

**Ex 9.3 Class 12 Maths Question 3.**

y = ae^{3x}+be^{-2x}

**Solution:**

Given that

y = ae^{3x}+be^{-2x} …(i)

**Ex 9.3 Class 12 Maths Question 4.**

y = e^{2x} (a+bx)

**Solution:**

y = e^{2x} (a+bx)

**Ex 9.3 Class 12 Maths Question 5.**

y = e^{x}(a cosx+b sinx)

**Solution:**

The curve y = e^{x}(a cosx+b sinx) …(i)

differentiating w.r.t x

**Ex 9.3 Class 12 Maths 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

**Ex 9.3 Class 12 Maths 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

**Ex 9.3 Class 12 Maths Question 8.**

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

**Solution:**

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

**Ex 9.3 Class 12 Maths Question 9.**

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

**Solution:**

Equation of the hyperbola is \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\)

Differentiating both sides w.r.t x

which is the req. differential eq. of the hyperbola.

**Ex 9.3 Class 12 Maths Question 10.**

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

**Solution:**

Let centre be (0, a) and r = 3

Equation of circle is

x² + (y – a)² = 9 …(i)

Differentiating both sides, we get

which is required equation

**Ex 9.3 Class 12 Maths Question 11.**

Which of the following differential equation has \(y={ c }_{ 1 }{ e }^{ x }+{ c }_{ 2 }{ e }^{ -x }\) as the general solution ?

(a) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0\)

(b) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

(c) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +1=0\)

(d) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -1=0\)

**Solution:**

(b) \(y={ c }_{ 1 }{ e }^{ x }+{ c }_{ 2 }{ e }^{ -x }\Rightarrow \frac { dy }{ dx } ={ c }_{ 1 }{ e }^{ x }-{ c }_{ 2 }{ e }^{ -x }\)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ c }_{ 1 }{ e }^{ x }+{ c }_{ 2 }{ e }^{ -x }\Rightarrow \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

**Ex 9.3 Class 12 Maths Question 12.**

Which of the following differential equations has y = x as one of its particular solution ?

(a) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }\frac { dy }{ dx } +xy=x\)

(b) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ x }\frac { dy }{ dx } +xy=x\)

(c) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }\frac { dy }{ dx } +xy=0\)

(d) \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ x }\frac { dy }{ dx } +xy=0\)

**Solution:**

(c) y = x

\(\frac { dy }{ dx } =1,\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0\)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }\frac { dy }{ dx } +xy=0\)

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