求微分方程dy/dx=(x+y)^2+3的通解
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求微分方程dy/dx=(x+y)^2+3的通解
![求微分方程dy/dx=(x+y)^2+3的通解](/uploads/image/z/7309287-63-7.jpg?t=%E6%B1%82%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8Bdy%2Fdx%3D%28x%2By%29%5E2%2B3%E7%9A%84%E9%80%9A%E8%A7%A3)
令t=x+y,则dy/dx=dt/dx-1
代入原方程,化简得d(t/2)/[1+(t/2)²]=2dx
==>arctan(t/2)=2x+C (C是任意常数)
==>t
==>x+y=2tan(2x+C)
故原方程的通解是y=2tan(2x+C)-x.
代入原方程,化简得d(t/2)/[1+(t/2)²]=2dx
==>arctan(t/2)=2x+C (C是任意常数)
==>t
==>x+y=2tan(2x+C)
故原方程的通解是y=2tan(2x+C)-x.