求齐次方程(1-2(x/y)^0.5)y'=1的通解y=[C/│u+2sqrt(u)-1│]*[sqrt(u)/(sqrt(u)+2)]^(√2/2)算到这步后 化不到正确答案x+2ye^(y/x)=c
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![求齐次方程(1-2(x/y)^0.5)y'=1的通解y=[C/│u+2sqrt(u)-1│]*[sqrt(u)/(sqrt(u)+2)]^(√2/2)算到这步后 化不到正确答案x+2ye^(y/x)=c](/uploads/image/z/7233917-5-7.jpg?t=%E6%B1%82%E9%BD%90%E6%AC%A1%E6%96%B9%E7%A8%8B%281-2%28x%2Fy%29%5E0.5%29y%27%3D1%E7%9A%84%E9%80%9A%E8%A7%A3y%3D%5BC%2F%E2%94%82u%2B2sqrt%28u%29-1%E2%94%82%5D%2A%5Bsqrt%28u%29%2F%28sqrt%28u%29%2B2%29%5D%5E%28%E2%88%9A2%2F2%29%E7%AE%97%E5%88%B0%E8%BF%99%E6%AD%A5%E5%90%8E+%E5%8C%96%E4%B8%8D%E5%88%B0%E6%AD%A3%E7%A1%AE%E7%AD%94%E6%A1%88x%2B2ye%5E%28y%2Fx%29%3Dc)
求齐次方程(1-2(x/y)^0.5)y'=1的通解y=[C/│u+2sqrt(u)-1│]*[sqrt(u)/(sqrt(u)+2)]^(√2/2)算到这步后 化不到正确答案x+2ye^(y/x)=c
求齐次方程(1-2(x/y)^0.5)y'=1的通解
y=[C/│u+2sqrt(u)-1│]*[sqrt(u)/(sqrt(u)+2)]^(√2/2)
算到这步后 化不到正确答案
x+2ye^(y/x)=c
求齐次方程(1-2(x/y)^0.5)y'=1的通解y=[C/│u+2sqrt(u)-1│]*[sqrt(u)/(sqrt(u)+2)]^(√2/2)算到这步后 化不到正确答案x+2ye^(y/x)=c
方程改写为dx/dy=1-2(x/y)^(1/2)
令x/y=u,则x=uy,dx/dy=ydu/dy+u代入上式得ydu/dy+u=1-2u^(1/2),该方程可以分离变量,即du/(1-2√u-u)=(1/y)dy
两边积分,∫du/(1-2√u-u)du=∫dy/y=ln│y│
令t=√u+1,则u=(t-1)^2,du=2(t-1)dt
计算∫du/(1-2√u-u)du=-∫du/[√u+1)^2-2]=-∫2(t-1)dt/(t^2-2)
=-∫2tdt/(t^2-2)+2∫dt/(t^2-2)
=-ln│t^2-2│+(1/√2)ln│(t-√2)/(t+√2)│
=ln│y│ +C1
把t=√u+1代回即可解出y=[C/│u+2sqrt(u)-1│]*[sqrt(u)/(sqrt(u)+2)]^(√2/2),C为常数,最后将u=x/y代回即可得出一个隐式通解f(x,y)=0来.
可以验证一下答案x+2ye^(y/x)=c是否正确。 解答见插图。