第 12 类 RD Sharma 解决方案 – 第 22 章微分方程 – 练习 22.5 |设置 2

问题 14. sin ⁴ x(dy/dx) = cosx

解决方案：

We have,

sin⁴x(dy/dx) = cosx          

dy = (cosx/sin⁴x)dx

Let, sinx = z

On differentiating both sides, we get 

cosx dx = dz

dy = (dz/z⁴)

On integrating both sides, we get

∫(dy) = ∫(1/z⁴)dz

y = (1/ -3t³) + c

y = -(1/3sin³x) + c

y = (-cosec³x/3) + c          -(Here, ‘c’ is integration constant)

问题 15. cosx(dy/dx) – cos2x = cos3x

解决方案：

We have,

cosx(dy/dx) – cos2x = cos3x     

(dy/dx) = (cos3x + cos2x)/cosx

(dy/dx) = (4cos³x – 3cosx + 2cos²x – 1)/cosx

(dy/dx) = (4cos³x/cosx) – 3(cosx/cosx) + 2(cos²x/cosx) – secx

dy = [4cos²x – 3 + 2cosx – secx]dx

dy = [4{(cos2x + 1)/2} – 3 + 2cosx – secx]dx

On integrating both sides, we get

∫dy = ∫[2cos2x – 1 + 2cosx – secx]dx

y = sin2x – x + 2sinx – log|secx + tanx| + c          -(Here, ‘c’ is integration constant)

问题 16. √(1 – x ⁴ )(dy/dx) = xdx

解决方案：

We have,

 √(1 – x⁴)(dy/dx) = xdx           

Let, x² = z

On differentiating both sides, we get 

2xdx = dz

xdx = (dz/2)

√(1 – z²)dy = (dz/2)

dy = $\frac{dz}{2\sqrt{1-z^2}}$

On integrating both sides, we get

∫dy = ∫$\frac{dz}{2\sqrt{1-z^2}}$

y = (1/2)sin^-1(z) + c

y = (1/2)sin^-1(x²) + c          -(Here, ‘c’ is integration constant)

问题 17. √(a + x)(dy) + xdx = 0

解决方案：

We have,

√(a + x)(dy) + xdx = 0       

dy = $\frac{-x}{\sqrt{a + x}}$dx

Let, (x + a) = z²

On differentiating both sides, we get 

dx = 2zdz

(x + a) = z²

x = z² – a

dy = -2[(z² – a)/z]zdz

On integrating both sides, we get

∫dy = -2∫[(z² – a)/z]zdz

y = -(2/3)(z³) + 2az + c

y = -(2/3)(x + a)^3/2 + 2a√(x + a) + c          -(Here, ‘c’ is integration constant)

问题 18. (1 + x ² )(dy/dx) – x = 2tan ^-1 x

解决方案：

We have,

(1 + x²)(dy/dx) – x = 2tan^-1x      

(1 + x²)(dy/dx) = 2tan^-1x + x

dy/dx = $\frac{2tan^{-1}x + x}{1 + x^2}$

dy $=(\frac{2tan^{-1}x+x}{1+x^2})dx$

On integrating both sides, we get

$∫dy=∫(\frac{2tan^{-1}x+x}{1+x^2})dx$

y = I₁ + I₂

I₁ = ∫($\frac{2tan^{-1}x}{1 + x^2}$

Let, tan^-1x = z

On differentiating both sides, we get 

$\frac{dx}{1 + x^2}$ = dz

= ∫2zdx

= z²

I₁ = (tan^-1x)²

I₂ = ∫$\frac{xdx}{(1+x^2)}$

= (1/2)log|1 + x²|

y = (tan^-1x)² + 1/2log|1 + x²|+ c          -(Here, ‘c’ is integration constant)

问题 19. (dy/dx) = xlogx

解决方案：

We have,

(dy/dx) = xlogx             

dy = xlogxdx

On integrating both sides, we get

∫dy = ∫xlogxdx

y = log|x|∫xdx – ∫[$\frac{(logx)}{dx}$∫xdx]dx

y = (x²/2)log|x| – ∫(1/x)(x²/2)dx

y = (x²/2)log|x| – ∫(x/2)dx

y = (x²/2)log|x| – (x²/4) + c          -(Here, ‘c’ is integration constant)

问题 20. (dy/dx) = xe ^x – (5/2) + cos ² x

解决方案：

We have,

(dy/dx) = xe^x – (5/2) + cos²x          

dy = (xe^x – (5/2) + cos²x) dx

On integrating both sides, we get

∫dy = ∫xe^x dx – 5/2∫dx + ∫cos²x dx

y = ∫xe^x dx – 5/2∫dx + ∫(1 + cos2x)/2 dx

= ∫xe^x dx – 5/2∫dx + 1/2∫dx + 1/2∫cos2x dx

= ∫xe^x dx – 2∫dx + 1/2∫cos2x dx

= x∫e^x dx – ∫(1∫e^x dx)dx – 2x + sin2x/4 dx

= xe^x – e^x – 2x + 1/4sin2x + c

问题 21. (x ³ + x ² + x + 1)(dy/dx) = 2x ² + x

解决方案：

We have,

(x³ + x² + x + 1)(dy/dx) = 2x² + x                  

(dy/dx) = (2x² + x)/(x³ + x² + x + 1)

dy = $\frac{2x^2+x}{(x+1)(x^2+1)}dx$

On integrating both sides, we get

∫dy = ∫$\frac{2x^2+x}{(x+1)(x^2+1)}dx$

Let,

$\frac{2x^2+x}{(x+1)(x^2+1)}dx=\frac{A}{(x+1)}+\frac{Bx+C}{x^2+1}$

2x² + x = Ax² + A + Bx² + Bx + Cx + C

2x² + x = (A + B)x² + (B + C)x + (A + C)

On comparing the coefficients on both sides,

(A + B) = 2

(B + C) = 1

(A + C) = 0

After solving the equations,

A = (1/2)

B = (3/2)

C = -(1/2)

y = (1/2)∫(dx/(x + 1) + $∫\frac{\frac{3}{2}x-\frac{1}{2}}{x^2+1}dx$

y = (1/2)log(x + 1) + (3/4)∫$\frac{2x}{x^2 + 1}$dx – (1/2)∫$\frac{dx}{(x^2 + 1)}$

y = (1/2)log|x + 1| + (3/4)log|x² + 1| – (1/2)tan^-1x + c          -(Here, ‘c’ is integration constant)

问题 22. sin(dy/dx) = k, y(0) = 1

解决方案：

We have,

sin(dy/dx) = k,           

(dy/dx) = sin^-1(k)

dy = sin^-1(k)dx

On integrating both sides, we get

∫dy = sin^-1(k)∫dx

y = xsin^-1k + c              -(1)

Put x = 0, y = 1

1 = 0 + c

1 = c

On putting the value of c in equation(1)

y = xsin^-1k + 1

y – 1 = xsin^-1x

问题 23. e ^(dy/dx) = x + 1, y(0) = 3

解决方案：

We have,

e^(dy/dx) = x + 1        

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

dy = log(x + 1)dx

On integrating both sides, we get

∫dy = ∫log(x + 1)dx

y = log(x + 1)∫dx – ∫[$\frac{dlog(x+1)}{dx}$∫dx]dx

y = xlog(x + 1) – ∫[x/(x + 1)]dx

y = xlog(x + 1) – ∫1 - \frac{1}{(x+1)}dx

y = xlog(x + 1) – x + log(x + 1) + c

y = (x + 1)log(x + 1) – x + c         -(1)

Put, y = 3, x = 0 in equation(1)

3 = 0 + c

y = (x + 1)log(x + 1) – x + 3

问题 24. c'(x) = 2 + 0.15x, c(0) = 100

解决方案：

We have,

c'(x) = 2 + 0.15x               -(1)

On integrating both sides, we get

∫c'(x)dx = ∫(2 + 0.15x)dx

c(x) = 2x + 0.15(x²/2) + c               -(2)

Put, c(0) = 100, x = 0 in equation(2)

100 = 2(0) + 0 + c

c = 100

c(x) = 2x + 0.15(x²/2) + 100

问题 25. x(dy/dx) + 1 = 0, y(-1) = 0

解决方案：

We have,

x(dy/dx) + 1 = 0                

xdy = -dx

dy = -(dx/x)

On integrating both sides, we get

∫dy = -∫(dx/x)

y = -logx + c           -(1)

Put, y = 0, x = -1 in equation(1)

0 = 0 + c

c = 0

y = -log|x|

问题 26. x(x ² – 1)(dy/dx) = 1, y(2) = 0

解决方案：

We have,

x(x² – 1)(dy/dx) = 1             -(1)     

dy = dx/x(x + 1)(x – 1)

On integrating both sides, we get

∫dy = ∫dx/x(x + 1)(x – 1)

Let, 1/x(x + 1)(x – 1) = A/x + B/(x + 1) + C/(x – 1)

1 = A(x + 1)(x – 1) + B(x)(x – 1) + C(x)(x + 1)             -(2) 

Put, x = 0, -1, 1 respectively and simplify above equation, we get,

A = -1, B = (1/2), C = (1/2)

y = $-∫\frac{dx}{x}+∫\frac{1}{2}\frac{dx}{(x+1)}+∫\frac{1}{2}\frac{dx}{(x-1)}$

y = -logx + (1/2)log(x + 1) + (1/2)log(x – 1)

y = (1/2)log(1/x²) + (1/2)log(x + 1) + (1/2)log(x – 1) + c             -(3) 

Put, y = 0, x = 2 in equation(3)

0 = (1/2)log(1/4) + (1/2)log(3) + 0 + c

c = -(1/2)log(3/4)

y = (1/2)log[(x² – 1)/x²] – (1/2)log(3/4)

$y=\frac{1}{2}log[\frac{4}{3}\frac{(x^2-1)}{x^2}]$