求微分方程dy/dx=(1+y2)/(x2-1)y的通解.
将方程分离变量,得[y/(1+y2)]dy=[1/(x2-1)]dx两边取不定积分,得∫[y/(1+y2)]dy=∫[1/(x2-1)]dx积分得1/2ln(1+y2)=(1/2)ln[(x-1)/(x+1)]+(1/2)lnC,变形得y2=C(x-1)/(x+1)-1,故方程的通解为y2=C(x-1)/(x+1)-1.
求微分方程dy/dx=(1+y2)/(x2-1)y的通解.
将方程分离变量,得[y/(1+y2)]dy=[1/(x2-1)]dx两边取不定积分,得∫[y/(1+y2)]dy=∫[1/(x2-1)]dx积分得1/2ln(1+y2)=(1/2)ln[(x-1)/(x+1)]+(1/2)lnC,变形得y2=C(x-1)/(x+1)-1,故方程的通解为y2=C(x-1)/(x+1)-1.