Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 6 Differential Equations Ex 6.2 Questions and Answers.

Maharashtra State Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

Question 1.
Obtain the differential equation by eliminating the arbitrary constants from the following equations:
(i) x3 + y3 = 4ax
Solution:
x3 + y3 = 4ax ……..(1)
Differentiating both sides w.r.t. x, we get
3x2 + 3y2 \(\frac{d y}{d x}\) = 4a × 1
∴ 3x2 + 3y2 \(\frac{d y}{d x}\) = 4a
Substituting the value of 4a in (1), we get
x3 + y3 = (3x2 + 3y2 \(\frac{d y}{d x}\)) x
∴ x3 + y3 = 3x3 + 3xy2 \(\frac{d y}{d x}\)
∴ 2x3 + 3xy2 \(\frac{d y}{d x}\) – y3 = 0
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

(ii) Ax2 + By2 = 1
Solution:
Ax2 + By2 = 1
Differentiating both sides w.r.t. x, we get
A × 2x + B × 2y \(\frac{d y}{d x}\) = 0
∴ Ax + By \(\frac{d y}{d x}\) = 0 ……..(1)
Differentiating again w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ii)
Substituting the value of A in (1), we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ii).1
This is the required D.E.

Alternative Method:
Ax2 + By2 = 1 ……..(1)
Differentiating both sides w.r.t. x, we get
A × 2x + B × 2y \(\frac{d y}{d x}\) = 0
∴ Ax + By \(\frac{d y}{d x}\) = 0 ……….(2)
Differentiating again w.r.t. x, we get,
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ii).2
The equations (1), (2) and (3) are consistent in A and B.
∴ determinant of their consistency is zero.
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ii).3
This is the required D.E.

(iii) y = A cos(log x) + B sin(log x)
Solution:
y = A cos(log x) + B sin (log x) ……. (1)
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (iii)

(iv) y2 = (x + c)3
Solution:
y2 = (x + c)3
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (iv)
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

(v) y = Ae5x + Be-5x
Solution:
y = Ae5x + Be-5x ……….(1)
Differentiating twice w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (v)
This is the required D.E.

(vi) (y – a)2 = 4(x – b)
Solution:
(y – a)2 = 4(x – b)
Differentiating both sides w.r.t. x, we get
2(y – a) . \(\frac{d}{d x}\)(y – a) = 4 \(\frac{d}{d x}\)(x – b)
∴ 2(y – a) . (\(\frac{d y}{d x}\) – 0) = 4(1 – 0)
∴ 2(y – a) \(\frac{d y}{d x}\) = 4
∴ (y – a) \(\frac{d y}{d x}\) = 2 ……..(1)
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (vi)
This is the required D.E.

(vii) y = a + \(\frac{a}{x}\)
Solution:
y = a + \(\frac{a}{x}\)
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (vii)
Substituting the value of a in (1), we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (vii).1
This is the required D.E.

(viii) y = c1e2x + c2e5x
Solution:
y = c1e2x + c2e5x ………(1)
Differentiating twice w.r.t. x, we get
\(\frac{d y}{d x}\) = c1e2x × 2 + c2e5x × 5
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (viii)
The equations (1), (2) and (3) are consistent in c1e2x and c2e5x
∴ determinant of their consistency is zero.
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (viii).1
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

Alternative Method:
y = c1e2x + c2e5x
Dividing both sides by e5x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (viii).2
This is the required D.E.

(ix) c1x3 + c2y2 = 5.
Solution:
c1x3 + c2y2 = 5 ……….(1)
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ix)
Differentiating again w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ix).1
The equations (1), (2) and (3) in c1, c2 are consistent.
∴ determinant of their consistency is zero.
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (ix).2
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

(x) y = e-2x(A cos x + B sin x)
Solution:
y = e-2x(A cos x + B sin x)
∴ e2x . y = A cos x + B sin x ………(1)
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (x)
Differentiating again w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (x).1
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q1 (x).2
This is the required D.E.

Question 2.
Form the differential equation of family of lines having intercepts a and b on the coordinate axes respectively.
Solution:
The equation of the line having intercepts a and b on the coordinate axes respectively, is
\(\frac{x}{a}+\frac{y}{b}=1\) ……….(1)
where a and b are arbitrary constants.
[For different values of a and b, we get, different lines. Hence (1) is the equation of family of lines.]
Differentiating (1) w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q2
Differentiating again w.r.t. x, we get \(\frac{d^{2} y}{d x^{2}}=0\)
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

Question 3.
Find the differential equation all parabolas having length of latus rectum 4a and axis is parallel to the X-axis.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q3
Let A(h, k) be the vertex of the parabola whose length of latus rectum is 4a.
Then the equation of the parabola is (y – k)2 = 4a (x – h), where h and k are arbitrary constants.
Differentiating w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q3.1
Differentiating again w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q3.2
This is the required D.E.

Question 4.
Find the differential equation of the ellipse whose major axis is twice its minor axis.
Solution:
Let 2a and 2b be lengths of major axis and minor axis of the ellipse.
Then 2a = 2(2b)
∴ a = 2b
∴ equation of the ellipse is
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)
i.e., \(\frac{x^{2}}{(2 b)^{2}}+\frac{y^{2}}{b^{2}}=1\)
∴ \(\frac{x^{2}}{4 b^{2}}+\frac{y^{2}}{b^{2}}=1\)
∴ x2 + 4y2 = 4b2
Differentiating w.r.t. x, we get
2x + 4 × 2y \(\frac{d y}{d x}\) = 0
∴ x + 4y \(\frac{d y}{d x}\) = 0
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

Question 5.
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Solution:
The equation of the line parallel to the line 2x + 3y + 4 = 0 is 2x + 3y + c = 0, where c is an arbitrary constant.
Differentiating w.r.t. x, we get
2 × 1 + 3 \(\frac{d y}{d x}\) + 0 = 0
∴ 3 \(\frac{d y}{d x}\) + 2 = 0
This is the required D.E.

Question 6.
Find the differential equation of all circles having radius 9 and centre at point (h, k).
Solution:
Equation of the circle having radius 9 and centre at point (h, k) is
(x – h)2 + (y – k)2 = 81 …… (1)
where h and k are arbitrary constant.
Differentiating (1) w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q6
Differentiating again w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q6.1
From (2), x – h = -(y – k) \(\frac{d y}{d x}\)
Substituting the value of (x – h) in (1), we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q6.2
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q6.3
This is the required D.E.

Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2

Question 7.
Form the differential equation of all parabolas whose axis is the X-axis.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q7
The equation of the parbola whose axis is the X-axis is
y2 = 4a(x – h) …… (1)
where a and h are arbitrary constants.
Differentiating (1) w.r.t. x, we get
2y \(\frac{d y}{d x}\) = 4a(1 – 0)
∴ y \(\frac{d y}{d x}\) = 2a
Differentiating again w.r.t. x, we get
Maharashtra Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.2 Q7.1
This is the required D.E.