Question

微积分：求下列微分方程的通解

如图，要有过程，求会做的解答

Answer 1

(a) dy/dx = 2xy, dy/y = 2xdx, lny = x^2 + lnC, y = Ce^(x^2)
(b) 一阶线性微分方程，
y = e^(∫xdx) [ ∫2xe^(-∫xdx)dx + C]
= e^(x^2/2)[∫2xe^(-x^2/2)dx + C]
= e^(x^2/2)[-2∫e^(-x^2/2)d(-x^2/2) + C]
= e^(x^2/2)[-2e^(-x^2/2) + C]
= -2 + Ce^(x^2/2)

Answer 2

(a)
dy/dx-2xy =0
dy/dx = 2xy
∫dy/y = ∫2x dx
ln|y| = x^2 +C'
y = C.e^(x^2)
(b)
dy/dx -xy =2x
dy/dx = x(y+2)
∫dy/(y+2)= ∫xdx
ln|y+2| = (1/2)x^2 + C'
y+2 = Ce^[(1/2)x^2]
y=-2 +Ce^[(1/2)x^2]