Separation of variables: dy/dx = 4x + y?

This is not a separable differential equation so it cannot be solved that way.

Rewrite this differential equation in standard form:

dy / dx = 4x + y

dy / dx - y = 4x

Find the complementary function by solving the auxiliary equation:

dy / dx - y = 0

m - 1 = 0

m = 1

yᶜ = C℮˟

Find the particular integral by comparing coefficients:

yᵖ = Ax + B

dyᵖ / dx = A

dyᵖ / dx - yᵖ = 4x

A - (Ax + B) = 4x

A - Ax - B = 4x

(A - B) - Ax = 4x

-A = 4

A = -4

A - B = 0

B = A

B = -4

yᵖ = -4x - 4

Find the general solution by combining these two parts:

y = yᶜ + yᵖ

y = C℮˟ - 4x - 4

You can't. It's a first order differential equation of the format

dy/dx + Py = Q

You need to multiply throughout by the integrating factor which is e to the power of the integral of Pdx.

In this case: You would leave the variables on the right side.

dy/dx = 4x + y

dy = (4x + y)dx

Integrate

y = 2x^2 + yx + C

if you are trying to solve diff equation then it is not question of variable separable. try using I.F.

multiply both side by e^(-x)

then your equation will come into

d(e^(-x)y)/dx=4xe^(-x) {d(e^(-x)y)/dx=yd(e^(-x))/dx+e^(-x)dy/dx}

which can be solved to

y=4[-x + 1] +ce^(x)

Where c is variable. it is first order diff equation.