Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 6 Differential Equations Ex 6.4 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 6 Differential Equations Ex 6.4

I. Solve the following differential equations:

Question 1.

\(x \sin \left(\frac{y}{x}\right) d y=\left[y \sin \left(\frac{y}{x}\right)-x\right] d x\)

Solution:

Question 2.

(x^{2} + y^{2}) dx – 2xy . dy = 0

Solution:

(x^{2} + y^{2}) dx – 2xy dy = 0

∴ 2xy dy = (x^{2} + y^{2}) dx

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

Question 3.

\(\left(1+2 e^{\frac{x}{y}}\right)+2 e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) \frac{d y}{d x}=0\)

Solution:

Question 4.

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

Question 5.

(x^{2} – y^{2}) dx + 2xy dy = 0

Solution:

Question 6.

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

Solution:

Question 7.

\(x \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0\)

Solution:

Question 8.

\(\left(1+e^{\frac{x}{y}}\right) d x+e^{\frac{x}{y}}\left(1-\frac{X}{y}\right) d y=0\)

Solution:

Question 9.

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

Solution:

Question 10.

xy \(\frac{d y}{d x}\) = x^{2} + 2y^{2}, y(1) = 0

Solution:

Question 11.

x dy + 2y · dx = 0, when x = 2, y = 1

Solution:

∴ x dy + 2y · dx = 0

∴ x dy = -2y dx

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

Integrating, we get

This is the general solution.

When x = 2, y = 1, we get

4(1) = c

∴ c = 4

∴ the particular solution is x^{2}y = 4.

Question 12.

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

Solution:

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

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

Put y = vx

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

Question 13.

(9x + 5y) dy + (15x + 11y) dx = 0

Solution:

(9x + 5y) dy + (15x + 11y) dx = 0

∴ (9x + 5y) dy = -(15x + 11y) dx

∴ \(\frac{d y}{d x}=\frac{-(15 x+11 y)}{9 x+5 y}\) ………(1)

Put y = vx

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

Question 14.

(x^{2} + 3xy + y^{2}) dx – x^{2} dy = 0

Solution:

(x^{2} + 3xy + y^{2}) dx – x^{2} dy = 0

∴ x^{2} dy = (x^{2} + 3xy + y^{2}) dx

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

Question 15.

(x^{2} + y^{2}) dx – 2xy dy = 0.

Solution: