12th Commerce Maths 1 Chapter 8 Exercise 8.2 Answers Maharashtra Board

Differential Equation and Applications Class 12 Commerce Maths 1 Chapter 8 Exercise 8.2 Answers Maharashtra Board

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 8 Differential Equation and Applications Ex 8.2 Questions and Answers.

Std 12 Maths 1 Exercise 8.2 Solutions Commerce Maths

Question 1.
Obtain the differential equation by eliminating arbitrary constants from the following equations:
(i) y = Ae ^3x + Be ^-3x
Solution:
y = Ae ^3x + Be ^-3x ……(1)
Differentiating twice w.r.t. x, we get

This is the required D.E.

(ii) y = \(c_{2}+\frac{c_{1}}{x}\)
Solution:
y = \(c_{2}+\frac{c_{1}}{x}\)
∴ xy = c ₂ x + c ₁
Differentiating w.r.t. x, we get

(iii) y = (c ₁ + c ₂ x) e ^x
Solution:
y = (c ₁ + c ₂ x) e ^x


This is the required D.E.

(iv) y = c ₁ e ^3x + c ₂ e ^2x
Solution:



This is the required D.E.

(v) y ² = (x + c) ³
Solution:
y ² = (x + c) ³
Differentiating w.r.t. x, we get

This is the required D.E.

Question 2.
Find the differential equation by eliminating arbitrary constant from the relation x ² + y ² = 2ax.
Solution:
x ² + y ² = 2ax
Differentiating both sides w.r.t. x, we get
2x + 2y\(\frac{d y}{d x}\) = 2a
Substituting value of 2a in equation (1), we get
x ² + y ² = [2x + 2y \(\frac{d y}{d x}\)]x = 2x ² + 2xy \(\frac{d y}{d x}\)
∴ 2xy \(\frac{d y}{d x}\) = y ² – x ² is the required D.E.

Question 3.
Form the differential equation by eliminating arbitrary constants from the relation bx + ay = ab.
Solution:
bx + ay = ab
∴ ay = -bx + ab
∴ y = \(-\frac{b}{a} x+b\)
Differentiating w.r.t. x, we get
\(\frac{d y}{d x}=-\frac{b}{a} \times 1+0=-\frac{b}{a}\)
Differentiating again w.r.t. x, we get
\(\frac{d^{2} y}{d x^{2}}\) = 0 is the required D.E.

Question 4.
Find the differential equation whose general solution is x ³ + y ³ = 35ax.
Solution:

Question 5.
Form the differential equation from the relation x ² + 4y ² = 4b ² .
Sol ution:
x ² + 4y ² = 4b ²
Differentiating w.r.t. x, we get
2x + 4(2y\(\frac{d y}{d x}\)) = 0
i.e. x + 4y\(\frac{d y}{d x}\) = 0 is the required D.E.