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

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2

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.