Question

Find all solutions of the given differential equations and then find the particular solutions for which a point (x,y) is given:

dy/dx = -32x + 8 ; (x,y) = (0,1)

Answer 1

$$\dfrac{dy}{dx}=-32x+8$$

$$dy=(-32x+8)dx$$

Integrate

$$\int dy=\int(-32x+8)dx$$

$$y=-16x^2+8x+C$$

The general solution of the given differential equation is

$$y(x)=-16x^2+8x+C$$

Point $(x,y)=(0,1)$

$$1=-16(0)^2+8(0)+C=>C=1$$

The particular solution is

$$y=-16x^2+8x+1$$