# Class 12 RD Sharma Solutions – Chapter 22 Differential Equations – Exercise 22.8

**Solve the following:**

**Question 1: **dy/dx = (x + y + 1)^{2}

**Solution:**

We have,

dy/dx = (x + y + 1)

^{2}Putting x + y + 1 = v

Therefore, dv/dx – 1 = v

^{2}⇒ dv/dx = v

^{2 }+ 1⇒ 1/(v

^{2 }+ 1) dv = dxIntegrating both sides, we get

∫ 1/(v

^{2 }+ 1) dv = ∫ dx

**Question 2: dy/dx cos (x – y) =** **1**

**Solution: **

We have,

dy/dx cos (x – y) = 1

⇒ dy/dx = 1/cos(x – y)

Putting x – y = v

⇒ 1 – dy/dx = dv/dx

⇒ dy/dx = 1 – dv/dx

Therefore, 1 – dv/dx = 1/cos v

⇒ dv / dx = 1 – 1/cos v

⇒ dv/dx = (cos v – 1)/cos v

⇒ cos v/ (cos v – 1) dv = dx

Integrating both sides, we get

∫cos v/(cos v – 1) dv = ∫dx

⇒ -∫(cos v (1 + cosv)) / (1 – cos

^{2}v) dv =∫ dx⇒ -∫(cos v (1 + cos v)) / (sin

^{2}v) dv = ∫ dx⇒ -∫(cot v cosec v + cot

^{2}v) dv = ∫ dx⇒ -∫ (cot v cosec v + cosec

^{2}v – 1) dv = ∫ dx⇒ -(-cosec v – cot v – v)= x + C

⇒ cosec ( x – y ) + cot ( x – y ) + x – y = x + C

⇒ cosec ( x – y ) + cot ( x – y ) – y = C

⇒ ((1+cos ( x – y )) / sin ( x – y )) – y = C

⇒ cot (( x – y )/ 2) = y + C

**Question 3: dy/dx = ((x – y) + 3)/ (2(x – y) + 5) **

**Solution: **

We have,

dy/dx = ((x – y) + 3)/ (2(x – y) + 5)

Putting x – y = v

⇒ 1 – dy/dx = dv/dx

⇒ dy/dx = 1 – dv/dx

Therefore, 1 – dv/dx = (v + 3)/ (2v + 5)

⇒ dv/dx = 1 – (v + 3)/ (2v + 5)

⇒ dv/dx = (2v + 5 – v – 3)/ 2v + 5

⇒ dv/dx = (v + 2) / (2v + 5)

⇒ (2v + 5)/(v + 2)dv = dx

Integrating both sides, we get

∫(2v + 5)/(v + 2) dv = ∫dx

⇒ ∫(2v + 4 + 1)/(v + 2) dv = ∫dx

⇒ ∫((2v + 4)/(v + 2) + 1/(v + 2))dv = ∫dx

⇒ 2∫dv + ∫1/(v + 2)dv = ∫dx

⇒ 2v + log |v + 2| = x + C

⇒ 2(x – y) + log | x – y + 2 | = x + C

**Question 4: dy/dx = (x + y)**^{2}

^{2}

**Solution: **

We have,

dy/dx = (x + y)

^{2}Let x + y = v

⇒ 1 + dy/dx = dv/dx

⇒ dy/dx = dv/dx – 1

Therefore, dv/dx – 1 = v

^{2}⇒ dv/dx = v

^{2}+ 1⇒ 1/(v

^{2}+ 1) dv = dxIntegrating both sides, we get

∫1/(v

^{2}+ 1) dv = ∫dx⇒ tan

^{-1}v = x + C⇒ v = tan (x + C)

⇒ x + y = tan (x + C)

**Question 5: (x + y)**^{2} dy/dx = 1

^{2}dy/dx = 1

**Solution: **

We have,

(x + y)

^{2}dy/dx = 1⇒ dy/dx = 1/( x + y)

^{2}Let x + y = v

⇒ 1 + dy/dx = dv/dx

⇒ dy/dx = dv/dx – 1

Therefore, dv/dx – 1 = 1/v

^{2}⇒ dv/dx = 1/v

^{2}+ 1⇒ v

^{2}/(v^{2}+ 1) dv = dxIntegrating both sides, we get

∫v

^{2}/(v^{2}+ 1) dv = ∫dx⇒ ∫v

^{2}+ 1 – 1/(v^{2}+ 1) dv = ∫dx⇒ ∫(1- 1/(v

^{2}+ 1) dv = ∫dx⇒ v – tan

^{-1}v = x + C⇒ x + y – tan

^{-1 }(x + y) = x + C⇒ y – tan

^{-1}(x + y) = C

**Question 6: cos**^{2} ( x – 2y) = 1 – 2dy/dx

^{2}( x – 2y) = 1 – 2dy/dx

**Solution: **

We have,

cos

^{2}( x – 2y ) = 1 – 2dy/dx⇒ 2dy/dx = 1 – cos

^{2}(x – 2y)Let x – 2y = v

⇒ 1 – 2 dy/dx = dv/dx

⇒ 2 dy/dx = 1 – dv/dx

Therefore, 1 – dv/dx = 1 – cos

^{2}v⇒ dv/dx = cos

^{2}v⇒ sec

^{2}v dv = dxIntegrating both sides, we get

∫ sec

^{2}v dv = ∫dx⇒ tan v = x – C

⇒ tan (x – 2y) = x – C

⇒ x = tan (x – 2y) + C

**Question 7: dy/dx = sec(x + y)**

**Solution: **

We have,

dy/dx = sec(x + y)

⇒ dy/dx = 1/cos ( x + y)

Let x + y = v

⇒ 1 + dy/dx = dv/dx

⇒ dy/dx = dv/dx -1

Therefore, dv/dx – 1 = 1/cos v

⇒ dv/dx = (cos v + 1)/ cos v

⇒ cos v/(cos v + 1) dv = dx

Integrating both sides, we get

∫ cos v/(cos v + 1) dv = ∫ dx

⇒ ∫ cos v (1 – cos v)/(1 – cos

^{2}v ) dv = ∫ dx⇒ ∫ cos v (1 – cos v)/sin

^{2}v dv = ∫ dx⇒ ∫ (cos v – cos^2 v)/sin

^{2}v dv = ∫ dx⇒ ∫(cot v cosec v – cot

^{2}v) dv = ∫ dx⇒ ∫(cot v cosec v – cosec

^{2}v + 1) dv = ∫ dx⇒ – cosec v + cot v + v = x + C

⇒ – cosec (x + y) + cot (x + y) + x + y = x + C

⇒ – cosec (x + y) + cot (x + y) + y = C

⇒ ((-1 + cos (x + y)) / sin (x + y)) + y = C

⇒ – tan ((x + y)/2) + y = C

⇒ y = tan((x + y)/2) + C

**Question 8: dy/dx = tan (x + y)**

**Solution: **

We have,

dy/dx = tan (x + y)

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

Let x + y =v

Therefore, 1 + dy/dx = dv/dx

⇒ dy/dx = dv/dx – 1

Since, dv/dx -1 = sin v/cos v

⇒ dy/dx = sin v/cos v + 1

⇒ dy/dx = (sin v + cos v)/ cos v

⇒ cos v/(sin v + cos v) dv = dx

Integrating both sides, we get

⇒ ∫ cos v/(sin v + cos v) dv =∫ dx

⇒ 1/2 ∫ {(sin v + cos v) + (cos v – sin v)}/(sin v + cos v) dv = ∫dx

⇒ 1/2 ∫ dv + 1/2 ∫ (cos v – sin v) / (sin v + cos v) dv = ∫dx

⇒ 1/2 v + 1/2 ∫ (cos v – sin v)/(sin v + cos v) dv = x

Putting sin v + cos v = t

⇒ (cos v – sin v) dv = dt

Therefore, 1/2 v + 1/2 ∫ dt/t = x

⇒ 1/2 v + 1/2 log |t| = x + C

⇒ (x + y) + 1/2 log |sin (x + y) + cos (x + y)| = x + C

⇒ 1/2 (y – x) + 1/2 log |sin (x + y) + cos (x + y)| = C

⇒ (y – x) + log |sin (x + y) + cos (x + y)| = 2C

⇒ y – x + log |sin (x + y) + cos (x + y)| = K (where K = 2C)

**Question 9: (x + y) (dx – dy) = dx + dy**

**Solution: **

We have,

(x + y) (dx – dy) = dx + dy

⇒ x dx + y dx -x dy – y dy = dx + dy

⇒ (x + y -1)dx = ( x + y +1) dy

⇒ dy/dx = (x + y -1)/(x + y + 1)

Let x + y = v

Therefore, 1 + dy/dx =dv/dx

⇒ dy/dx = dv/dx – 1

Therefore, dv/dx -1 = (v – 1)/(v +1)

⇒ dv/dx = (v – 1)/(v +1) + 1

⇒ dv/dx = (v – 1 + v +1)/(v +1)

⇒ dv/dx = 2v / (v +1)

⇒ (v +1) / 2v dv = dx

Integrating both sides, we get

∫(v + 1)/2v dv = ∫ dx

⇒ 1/2 ∫dv + 1/2 ∫1/v dv = ∫dx

⇒ 1/2 v + 1/2 log |v| = x + C

⇒ 1/2 (x + y) + 1/2 log |x + y| = x + C

⇒ 1/2 (y – x) + 1/2 log |x + y| = C

**Question 10: (x + y + 1)dy/dx = 1**

**Solution: **

We have,

(x + y + 1)dy/dx =1

⇒ dy/dx = 1/(x + y + 1)

Let x + y + 1 = v

Therefore, 1 + dy/dx = dv/dx

⇒ dy/dx = dv/dx – 1

Therefore, dv/dx – 1= 1/v

⇒ dv/dx = 1/v + 1

⇒ v/(v + 1) dv = dx

Integrating both sides, we get

∫ v/(v + 1) dv = ∫ dx

⇒ ∫ (v + 1 – 1)/(v + 1) dv = ∫ dx

⇒ ∫ (1 – 1/(v + 1))dv = ∫ dx

⇒ v – log |v + 1| = x + K

⇒ x + y + 1 – log |x + y+ 1 + 1| = x + K

⇒ y – log |x + y + 2| = K – 1

⇒ y – log |x + y + 2| = C

_{1}( C_{1}= K – 1)⇒ y – C

_{1}= log |x + y + 2|⇒ e

^{y – C}_{1}= x + y + 2⇒ e

^{y }/ e^{C}_{1}= x + y + 2⇒ e

^{– C}_{1 }e^{y}= x + y + 2⇒ C e

^{y}= x + y + 2 (C = e^{– C}_{1})⇒ x = C e

^{y}– y – 2

**Question 11: dy/dx + 1 = e**^{x + y }

^{x + y }

**Solution: **

We have,

dy/dx + 1 = e

^{x + y }. . . (1)Let x + y = t

⇒ 1 + dy/dx = dt/dx

Substituting the value of x + y = t and 1 + dy/dx = dt/dx from (1), we get

dt/dx = e

^{t}⇒ e

^{– t}dt = dx⇒ – e

^{– t}= x + C⇒ – e

^{– (x + y)}= x + C [Since t = x + y]

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