Question

Find the particular integral of dy/dx+y=cos3x

Answer 1

$$y'+y= cos(3x)$$

Consider:

$$y(x)=u(x)*v(x)\\ u'v+v'u+uv=cos(3x)\\ uv'+v(u'+u)=cos(3x)\\$$

then:

$$\frac{du}{dx}=-u\\ \frac{du}{u}=-dx\\ \int \frac{du}{u}=-\int dx\\ ln(u)=-x \\ u=e^{-x}$$

then:

$$v'u=cos(3x)\\ v'=cos(3x)*e^{x}\\ v=\int cos(3x)*e^x=\frac{1}{10}cos(3x)*e^x+\frac{3}{10}sin(3x)*e^x+C\\$$

then a particular integral is

$$y(x)=\frac{1}{10}cos(3x)+\frac{3}{10}sin(3x)$$