If the right side of the equation dy dx = f(x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous. such equations can always be transformed into separable equations by a change of the dependent variable. the following method outline can be used for any homogeneous equation. that is, the substitution y = xv(x) transforms a homogeneous equation into a separable equation. the latter equation can be solved by direct integration, and then replacing v by y x gives the solution to the original equation. dy/dx = (x^2 + 5y^2)/ 2xy

 y = x*sqrt(Cx - 1)

Step-by-step explanation:

Given:

                                  dy / dx = (x^2 + 5y^2) / 2xy

Find:

Solve the given ODE by using appropriate substitution.

Solution:

- Rewrite the given ODE:

                               dy/dx = 0.5(x/y) + 2.5(y/x)

- use substitution y = x*v(x)

                               dy/dx = v + x*dv/dx

- Combine the two equations:

                                v + x*dv/dx = 0.5*(1/v) + 2.5*v

                                x*dv/dx = 0.5*(1/v) + 1.5*v

                                x*dv/dx = (v^2 + 1) / 2v

-Separate variables:

                                 (2v.dv / (v^2 + 1) = dx / x  

- Integrate both sides:

                                 Ln (v^2 + 1) = Ln(x) + C

                                 v^2 + 1 = Cx

                                 v = sqrt(Cx - 1)

- Back substitution:

                                (y/x) = sqrt(Cx - 1)

                               y = x*sqrt(Cx - 1)

9) i can't see the angle so whatever it is just do this: cos(angle) x hypotenuse and then use that answer and they hypotenuse and plug it into this: (answer you just got)^2 + b^2 = (hypotenuse)^2. the first answer will be the answer of the adjacent side and the second answer after this equation will be the opposite side.

10) y = 14 x = √147 or 12.12435565

11) x = 4.8

12) x = 4.2

13) x = 9.3

14) pq: 13.2

qr : 17.6

angle p: 53º

15) angle c: 54.3º

angle a: 35.7º

16) h = 21040

v = 1841

step-by-step explanation:

10) 7/sin(30) = 14 = y

7^2 + b^2 = 14^2

49 + b^2 = 196

b^2 = 196 - 49

√b^2 = √147

x = √147

11) cos(70) x 14 = 4.788282007

x = 4.8

12) tan(25) x 9 = 4.196768923

x = 4.2

13) sin(31) x 18 = 9.270685348

x = 9.3

14) pq: sin(37) x 22 = 13.23993051

pq: 13.2

qr: cos(37) x 22 = 17.56998122

qr : 17.6

angle p: 180 - (37 + 90) = 53º

15) cos-1(7/12) = 54.31466529

angle c: 54.3º

angle a: 180 - (90 + 54.31466529) = 35.68533471

angle a: 35.7º

16) 4 x 5280 = 21120

cos(5) x 21120

h = 21039.63202

h = 21040

sin(5) x 21120

v = 1840.729287

v = 1841

it would take 7/10 of an hour too fill the pool.

step-by-step explanation: