如何计算微分方程的近似解？

Question

如题所述

Answer 1

1、∫(0,π/2) cos^6xdx
=∫(0,π/2) cos^5xd(sinx)
=cos^5xsinx|(0,π/2)+∫(0,π/2) 5sin^2xcos^4xdx
=∫(0,π/2) 5(1-cos^2x)cos^4xdx
=5∫(0,π/2) cos^4xdx-5∫(0,π/2) cos^6xdx
∫(0,π/2) cos^6xdx=(5/6)*∫(0,π/2) cos^4xdx
=(5/6)*(3/4)*∫(0,π/2) cos^2xdx
=(5/6)*(3/4)*(1/2)*∫(0,π/2) dx
=(5/6)*(3/4)*(1/2)*(π/2)
=5π/32
2、∫(0,1) f(x)/√xdx
=∫(0,1) (1/√x)dx∫(1,√x)e^(-t^2)dt
=-∫(0,1) (1/√x)dx∫(√x,1)e^(-t^2)dt
=-∫(0,1) e^(-t^2)dt∫(0,t^2)1/√xdx
=-∫(0,1) e^(-t^2)dt*2√x|(0,t^2)
=-∫(0,1) 2te^(-t^2)dt
=∫(0,1) e^(-t^2)d(-t^2)
=e^(-t^2)|(0,1)
=1/e-1