第12类RD Sharma解–第22章微分方程–练习22.8

解决以下问题：

问题1： dy / dx =(x + y + 1) ²

解决方案：

We have,

dy/dx = (x + y + 1)²

Putting x + y + 1 = v

Therefore, dv/dx – 1 = v²

⇒ dv/dx = v² + 1

⇒ 1/(v² + 1) dv = dx

Integrating both sides, we get

∫ 1/(v² + 1) dv = ∫ dx

问题2：dy / dx cos(x – y)= 1

解决方案：

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 – cos² v) dv =∫ dx

⇒ -∫(cos v (1 + cos v)) / (sin² v) dv = ∫ dx

⇒ -∫(cot v cosec v + cot² v) dv = ∫ dx

⇒ -∫ (cot v cosec v + cosec² 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

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

解决方案：

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

问题4：dy / dx =(x + y) ²

解决方案：

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 = v²

⇒ dv/dx = v² + 1

⇒ 1/(v² + 1) dv = dx

Integrating both sides, we get

∫1/(v² + 1) dv = ∫dx

⇒ tan^-1 v = x + C

⇒ v = tan (x + C)

⇒ x + y = tan (x + C)

问题5：(x + y) ² dy / dx = 1

解决方案：

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/v²

⇒ dv/dx = 1/v² + 1

⇒ v²/(v² + 1) dv = dx

Integrating both sides, we get

∫v²/(v² + 1) dv = ∫dx

⇒ ∫v² + 1 – 1/(v² + 1) dv = ∫dx

⇒ ∫(1- 1/(v² + 1) dv = ∫dx

⇒ v – tan^-1 v = x + C

⇒ x + y – tan^-1 (x + y) = x + C

⇒ y – tan^-1 (x + y) = C

问题6：cos ² (x – 2y)= 1 – 2dy / dx

解决方案：

We have,

cos² ( x – 2y ) = 1 – 2dy/dx 

⇒ 2dy/dx = 1 – cos² (x – 2y)

Let x – 2y = v

⇒ 1 – 2 dy/dx = dv/dx

⇒ 2 dy/dx = 1 – dv/dx

Therefore, 1 – dv/dx = 1 – cos² v

⇒ dv/dx = cos² v

⇒ sec² v dv = dx

Integrating both sides, we get

∫ sec² v dv = ∫dx

⇒ tan v = x – C

⇒ tan (x – 2y) = x – C

⇒ x = tan (x – 2y) + C

问题7：dy / dx = sec(x + y)

解决方案：

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 – cos² v ) dv = ∫ dx

⇒ ∫ cos v (1 – cos v)/sin² v  dv = ∫ dx

⇒ ∫ (cos v – cos^2 v)/sin² v  dv = ∫ dx

⇒ ∫(cot v cosec v – cot² v) dv = ∫ dx

⇒ ∫(cot v cosec v – cosec² 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

问题8：dy / dx = tan(x + y)

解决方案：

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)

问题9：(x + y)(dx-dy)= dx + dy

解决方案：

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

问题10：(x + y + 1)dy / dx = 1

解决方案：

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| = C₁        ( C₁ = K – 1)

⇒ y – C₁ = log |x + y + 2|

⇒ e^{y – C₁} = x + y + 2

⇒ e^y / e^C₁ = x + y + 2

⇒ e^{– C₁}  e^y = x + y + 2

⇒ C e^y  = x + y + 2                  (C = e^{– C₁})

⇒ x =  C e^y – y – 2  

问题11：dy / dx +1 = e ^{x + y}

解决方案：

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]