Find the general solution to the following differential equa

Solution

Let mu(x) = exp( integral 2/x dx) = x^2. Multiply both sides by mu(x): x^2 ( dy(x))/( dx)+(2 x) y(x) = sin(3 x) Substitute 2 x = ( d)/( dx)(x^2): x^2 ( dy(x))/( dx)+( d)/( dx)(x^2) y(x) = sin(3 x) Apply the reverse product rule f ( dg)/( dx)+( df)/( dx) g = ( d)/( dx)(f g) to the left-hand side: ( d)/( dx)(x^2 y(x)) = sin(3 x) Integrate both sides with respect to x: integral ( d)/( dx)(x^2 y(x)) dx = integral sin(3 x) dx Evaluate the integrals: x^2 y(x) = -1/3 cos(3 x)+c_1, where c_1 is an arbitrary constant. Divide both sides by mu(x) = x^2: y(x) = (-1/3 cos(3 x)+c_1)/x^2