Question

微分方程求解 (x^2y^3+xy)dy=dx

Answer 1

令z=1/x,则dx=-x²dz

代入原方程得 (x²y³+xy)dy=-x²dz

==>dz/dy+y/x=-y³

==>dz/dy+yz=-y³.(1)

∵方程(1)一阶线性微分方程

∴由一阶线性微分方程通解公式,得

方程(1)的通解是z=Ce^(y²/2)-y²+2 (C是积分常数)

==>1/x=Ce^(y²/2)-y²+2

==>[Ce^(y²/2)-y²+2]x=1

故原方程的通解是[Ce^(y²/2)-y²+2]x=1 (C是积分常数).