微分方程（y+x3）dx-2xdy=0满足y|x=1=65的特解为y＝x+15x3y＝x+15x3．

Question

微分方程（y+x³）dx-2xdy=0满足y|_x=1=(6)/(5)的特解为

y＝√(x)+(1)/(5)x3

y＝√(x)+(1)/(5)x3

．

Answer 1

(dy)/(dx)−(1)/(2x)y＝(1)/(2)x2
利用公式法求一阶线性方程通解
y＝e

∫

(1)/(2x)dx[

∫

(1)/(2)x2e−

∫

(1)/(2x)dx+C]=e(1)/(2)lnx[

∫

(1)/(2)x2e−

∫

(1)/(2x)dx+C]
=√(x)[

∫

(1)/(2)x(3)/(2)dx+C]=√(x)[(1)/(5)x(5)/(2)+C]
∵Y(1)＝(6)/(5)⇒C＝1
∴y＝√(x)+(1)/(5)x3．