Question

# Find the general solution of the first-order linear differential equation (dy/dx)+(1/x)y = 6x + 2, for x > 0

First order differential equations
Find the general solution of the first-order linear differential equation $$(dy/dx)+(1/x)y = 6x + 2$$, for x > 0

2021-01-17

$$(dy/dx)+(1/x)y = 6x + 2$$
Now, compare with $$dy/dx+Py=Q$$, we get
$$P=1/x\text{ and }Q= 6x+2$$
$$I.F. = e^{\int(1/x dx)}$$
$$= e^{(\ln x)}$$
= x
$$y*x= \int (6x+2)x dx$$
$$xy= \int 6x^2*dx+\int 2x*dx$$
$$xy= 2x^3+x^2+C$$, where C is a constant
$$dy/dx+(1/x)y= 6x+2\text{ is }xy= 2x^3+x^2+C$$