Step 3 Now we must multiply both sides of the given equation by the integrating factor e-2x. e-2x/ dy dx -2x dy dx -2y) = e-²x(x² + 3) -2x -2x 2ye = X e + By the choice of the integrating function and the chain rule, the left side of the equation can always be simplified as follows. el P(x) dx dy + P(x)e/P(x) dxy = d [@SP(x) dxy] dx dx Thus, our equation simplifies as the following. -2xy = x²e-2x₁ & [e-2xy] = + dx -2x -2x If the right side of the equation dependent variable. dy dx = f(x, y) can be expressed as a function of the ratio only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the 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 gives the solution to the original equation. X (x² + 3xy + y²) dx - x² dy = 0 (a) Show that the given equation is homogeneous. Dividing by x², we see that the equation becomes (b) Solve the differential equation. |y(x) = = -X log C(x) +1 X X 3y + 2 X )dx-dy. Hence the differential equation is homogeneous.