Q:

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

Accepted Solution

A:
Answer:[tex]y(x)=\frac{2*(cos(4x)+2-4*C_1)}{cos(4x)-4C_1}[/tex]Step-by-step explanation:Rewrite the equation as:[tex]\frac{dy(x)}{dx}=(y-2)^{2} *sin(4x)[/tex]   (1)divide both sides of (1) by [tex](y-2)^{2}[/tex] [tex]\frac{\frac{dy}{dx} }{(y-2)^{2} } =sin(4x)[/tex]Now integrate both sides:[tex]\int\ \frac{1}{(y-2)^{2} } } \, dy = \int\ sin(4x) } } dx[/tex]Solving the left side integral:Let:[tex]u=y-2\\du=dy[/tex]Replacing [tex]u[/tex] and [tex]du[/tex][tex]\int\ \frac{du}{u^{2} }  } }=-\frac{1}{u}[/tex][tex]u=y-2[/tex] then:[tex]-\frac{1}{y-2}[/tex]Solving the right side integral:[tex]\int\ sin(4x) } } dx=-\frac{1}{4} cos(4x)+C_1[/tex]Now we got this:[tex]-\frac{1}{y-2}=-\frac{1}{4} cos(4x)+C_1[/tex]Finally, solving for y:[tex]y=\frac{2*(cos(4x)+2-4*C_1)}{cos(4x)-4C_1}[/tex]