Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 8 Differential Equation and Applications Ex 8.4 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4

Solve the following differential equations:

Question 1.

x dx + 2y dy = 0

Solution:

x dx + 2y dy = 0

Integrating, we get

∫x dx + 2 ∫y dy = c_{1}

∴ \(\frac{x^{2}}{2}+2\left(\frac{y^{2}}{2}\right)=c_{1}\)

∴ x^{2} + 2y^{2} = c, where c = 2c_{1}

This is the general solution.

Question 2.

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

Solution:

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

∴ (xy + x^{2}) dy = -y^{2} dx

∴ \(\frac{d y}{d x}=\frac{-y^{2}}{x y+x^{2}}\) ………(1)

Put y = vx

∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)

Substituting these values in (1), we get

This is the general solution.

Question 3.

x^{2}y dx – (x^{3} + y^{3}) dy = 0

Solution:

x^{2}y dx – (x^{3} + y^{3}) dy = 0

∴ (x^{3} + y^{3}) dy = x^{2}y dx

∴ \(\frac{d y}{d x}=\frac{x^{2} y}{x^{3}+y^{3}}\) ……(1)

Put y = vx

∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)

This is the general solution.

Question 4.

\(\frac{d y}{d x}+\frac{x-2 y}{2 x-y}=0\)

Solution:

This is the general solution.

Question 5.

(x^{2} – y^{2}) dx + 2xy dy = 0

Solution:

(x^{2} – y^{2}) dx + 2xy dy = 0

∴ 2xy dy = -(x^{2} – y^{2}) dx = (y^{2} – x^{2}) dx

∴ \(\frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y}\) ………(1)

Question 6.

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

Solution:

Question 7.

x^{2}\(\frac{d y}{d x}\) = x^{2} + xy – y^{2}

Solution: