2.6.1—Linear differential equations

Write the equation as dy/dx+P(x)y=Q(x). The integrating factor is μ=e^{∫P(x)dx}; multiplying through makes the left side d(μy)/dx, so μy=∫μQ dx+C.

Put every y term on the left and divide by the coefficient of dy/dx before choosing μ. Apply an initial condition only after the general solution is obtained.

For dy/dx+2y=e^x, μ=e^{2x}; then d(e^{2x}y)/dx=e^{3x}, so y=(1/3)e^x+Ce^{−2x}.

The integrating factor multiplies the whole equation, not just y, and P must be read after standardising the derivative coefficient.