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2xy 4x

解微分方程:y'+2xy=4x 解:y'=(4-2y)x 分离变量得 dy/(4-2y)=xdx 取积分得 -(1/2)∫d(4-2y)/(4-2y)=∫xdx 积分之得 -(1/2)ln∣4-2y∣=(1/2)x²+(1/2)lnc 即 ln∣4-2y∣=x²+lnc 故∣4-2y∣=e^(x²+lnc)=ce^(x²) 4-2y=±ce^(x²) ...

dy/dx=4x-2xy=2x(2-y) dy/(2-y)=2xdx -ln(2-y)=x^2+C 再化简把y写成x、C的表达式即可。

方程左边恰好是一个导数 [(x²+1)y]'=4x²,两边积分得 (x²+1)y=∫4x²dx=4x³/3+C, 即所求微分方程的通解为 (x²+1)y=4x³/3+C.

∂f/∂x=3x²-8x+2y ∂f/∂y=2x-2y 驻点:(0,0)、(2,2) ∂²f/∂x²=6x-8=A ∂²f/∂x∂y=2=B ∂²f/∂y²=-2=C (0,0) B²-AC

分解因式

各项为5,-2xy和-4x三次方y,次数为4

(4x-2y-1)²+√(xy-2)=0 ∴{4x-2y-1=0 xy-2=0 即{2x-y=1/2 xy=2 4x²y-4x²y²-2xy² =2xy(2x-2xy-y) =2×2×[(2x-y)-2×2] =4×(1/2-4) =2-16 =-14

5x²y+2xy²+(-2xy²)+4x²y =5x²y+2xy²-2xy²+4x²y =9x²y 5x²y+(-2xy²)+(-2xy²)+4x²y =5x²y-2xy²-2xy²+4x²y =9x²y-4xy²

x+3=(1/8)(8x+4)+1∫(x+3)/(4x^2+4x+3)dx=(1/8)∫(8x+4)/(4x^2+4x+3)dx+∫dx/(4x^2+4x+3)=(1/8)ln|4x^2+4x+3|+∫dx/(4x^2+4x+3)=(1/8)ln|4x^2+4x+3|+√2arctan[(2x+1)/√2]+Cconsider4x^2+4x+3=(2x+1)^2+2let2x+1=√2tanu2dx=√2(secu)^2du∫dx/(4x^2+4x+...

供参考。

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