Please show a step by step process to this question Find all

Please show a step by step process to this question:

Find all solutions of the following linear differential equations, y\' + 2xy = 2x; y(0) = 1.

Solution

Solve for ( dy(x))/( dx): ( dy(x))/( dx) = -2 (-x+x y(x)) Simplify: ( dy(x))/( dx) = x (-2 y(x)+2) Divide both sides by -2 y(x)+2: (( dy(x))/( dx))/(-2 y(x)+2) = x Integrate both sides with respect to x: integral (( dy(x))/( dx))/(-2 y(x)+2) dx = integral x dx Evaluate the integrals: -1/2 log(-2 y(x)+2) = x^2/2+c_1, where c_1 is an arbitrary constant. Solve for y(x): y(x) = -1/2 e^(-x^2-2 c_1)+1 Simplify the arbitrary constant: y(x) = c_1 e^(-x^2)+1 Solve for c_1 using the initial conditions: Substitute y(0) = 1 into y(x) = c_1 e^(-x^2)+1: c_1+1 = 1 Solve the equation: c_1 = 0 Substitute c_1 = 0 into y(x) = c_1 e^(-x^2)+1: y(x) = 1