Question

Question #b8c93

Answer 1

`y = Ce^(-x^2) + 2`

Explanation:

Given: `dy/dx+2xy=4x`

Subtract `2xy` from both sides:

#dy/dx = 4x-2xy

Factor:

`dy/dx = -2x(y-2)`

This is a separable equation:

`dy/(y-2) = -2xdx`

After integration:

`ln|y-2| = -x^2+C`

Make both sides exponents of e:

`y -2 = Ce^(-x^2)`

Add 2 to both sides:

`y = Ce^(-x^2) + 2`

Answer 2

The solution is `y=2+C_1e^(-x^2)`

Explanation:

This is first order differential equation with separable variables

`dy/dx+2xy=4x`

`dy/dx=4x-2xy=-2x(y-2)`

`dy/(y-2)=-2xdx`

`intdy/(y-2)=-int2xdx`

`-ln(y-2)=-2x^2/2+C`

`ln(y-2)=-x^2+C`

`y-2=C_1e^(-x^2)`

`y=2+C_1e^(-x^2)`