Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 4 Differential Equations Ex 4.2 Text Book Back Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.2

Question 1.

Solve:

(i) \(\frac { dy }{dx}\) = ae^{y}

Solution:

\(\frac { dy }{dx}\) = ae^{y}

\(\frac { dy }{e^y}\) = adx ⇒ e^{-y} dy = adx

Integrating on both sides

∫e^{-y} dy = ∫adx

\(\frac { e^y }{(-1)}\) = ax + c

-e^{-y} = ax + c ⇒ e^{-y} + ax + c = 0

(ii) \(\frac { 1+x^2 }{1+y}\) = xy \(\frac { dy }{dx}\)

Solution:

Question 2.

y(1 – x) – x \(\frac { dy }{dx}\) = 0

Solution:

Question 3.

(i) ydx – xdy = 0 dy

Solution:

ydx – xdy = 0

ydx = xdy

\(\frac { 1 }{x}\) dx = \(\frac { 1 }{y}\) dy

Integrating on both sides

∫\(\frac { 1 }{x}\)dx = ∫\(\frac { 1 }{y}\)dy

log x = log y + log c

log x = log cy

⇒ x = cy

(ii) \(\frac { dy }{dx}\) + e^{x} + ye^{x} = 0

Solution:

Question 4.

Solve : cosx (1 + cosy) dx – siny (1 + sinx) dy = 0

Solution:

cos x (1 + cos y) dx – sin y (1 + sin x) dy = 0

cos x (1 + cos y) dx = sin y (1 + sin x) dy

Question 5.

Solve: (1 – x) dy – (1 + y) dx = 0

Solution:

(1 – x) dy – (1 + y) dx = 0

(1 – x) dy = (1 + y) dx

\(\frac { dy }{(1+y)}\) = \(\frac { dx }{(1-x)}\)

Integrating on both sides

Question 6.

Solve:

(i) \(\frac { dy }{dx}\) = y sin 2x

Solution:

(ii) log(\(\frac { dy }{dx}\)) = ax + by

Solution:

Question 7.

Find the curve whose gradient at any point P (x, y) on it is \(\frac { x-a }{y-b}\) and which passes through the origin.

Solution:

The gradient of the curve at P (x, y)

\(\frac { dy }{dx}\) = \(\frac { x-a }{y-b}\)

(y – b) dy = (x – a) dx

Integrating on both sides

∫(y – b)dy = ∫(x – a)dx

⇒ \(\frac { (y-b)^2 }{2}\) = \(\frac { (x-a)^2 }{2}\) + c

(Multiply each term by 2)

∴ (y – b)² = (x – a)² + 2c ……… (1)

Since the curve passes through the origin (0, 0)

eqn (1) (0 – b)² = (0 – a)² + 2c

b² = a² + 2c

b² – a² = 2c ………. (2)

Substitute eqn (2) in eqn (1)

(y – b)² = (x – a)² + b² – a²