Question

Using separation of variables, solve y′=(y−2)^2 sin(4x)

Answer 1

`y(x)=(2*(cos(4x)+2-4*C_1))/(cos(4x)-4C_1)`

Explanation:

Rewrite the equation as:

`(dy(x))/(dx)=(y-2)^(2) *sin(4x)` (1)

divide both sides of (1) by
`(y-2)^(2)`

`((dy)/(dx) )/((y-2)^(2) ) =sin(4x)`

Now integrate both sides:

`\int\ (1)/((y-2)^(2) ) } \, dy = \int\ sin(4x) } } dx`

Solving the left side integral:

Let:

`u=y-2\\du=dy`

Replacing
`u` and
`du`

`\int\ (du)/(u^(2) )  } }=-(1)/(u)`

`u=y-2` then:

`-(1)/(y-2)`

Solving the right side integral:

`\int\ sin(4x) } } dx=-(1)/(4) cos(4x)+C_1`

Now we got this:

`-(1)/(y-2)=-(1)/(4) cos(4x)+C_1`

Finally, solving for y:

`y=(2*(cos(4x)+2-4*C_1))/(cos(4x)-4C_1)`