如题所述
(4)解:∵y'-3xy=2x
==>e^(-3x^2/2)dy-3xye^(-3x^2/2)dx=2xe^(-3x^2/2)dx (等式两端同乘e^(-3x^2/2)dx)
==>d(ye^(-3x^2/2))=(-2/3)d(e^(-3x^2/2))
==>∫d(ye^(-3x^2/2))=(-2/3)∫d(e^(-3x^2/2))
==>ye^(-3x^2/2)=(-2/3)e^(-3x^2/2)+C (C是积分常数)
==>y=Ce^(3x^2/2)-2/3
∴此方程的通解是y=Ce^(3x^2/2)-2/3。
(8)解:∵xy'-y=2
==>(xdy-ydx)/x^2=2dx/x^2 (等式两端同乘dx/x^2)
==>d(y/x)=2dx/x^2
==>∫d(y/x)=2∫dx/x^2
==>y/x=C-2/x (C是积分常数)
==>y=Cx-2
∴此方程的通解是y=Cx-2
∵y(1)=0
∴代入通解得 C=2
故所求特解是y=2(x-1)。
说明:此两题也可以直接应用一阶线性微分方程通解公式求解。
==>e^(-3x^2/2)dy-3xye^(-3x^2/2)dx=2xe^(-3x^2/2)dx (等式两端同乘e^(-3x^2/2)dx)
==>d(ye^(-3x^2/2))=(-2/3)d(e^(-3x^2/2))
==>∫d(ye^(-3x^2/2))=(-2/3)∫d(e^(-3x^2/2))
==>ye^(-3x^2/2)=(-2/3)e^(-3x^2/2)+C (C是积分常数)
==>y=Ce^(3x^2/2)-2/3
∴此方程的通解是y=Ce^(3x^2/2)-2/3。
(8)解:∵xy'-y=2
==>(xdy-ydx)/x^2=2dx/x^2 (等式两端同乘dx/x^2)
==>d(y/x)=2dx/x^2
==>∫d(y/x)=2∫dx/x^2
==>y/x=C-2/x (C是积分常数)
==>y=Cx-2
∴此方程的通解是y=Cx-2
∵y(1)=0
∴代入通解得 C=2
故所求特解是y=2(x-1)。
说明:此两题也可以直接应用一阶线性微分方程通解公式求解。
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第1个回答 2015-10-05
答案如下:
(4)解:∵y'-3xy=2x
==>e^(-3x^2/2)dy-3xye^(-3x^2/2)dx=2xe^(-3x^2/2)dx (等式两端同乘e^(-3x^2/2)dx)
==>d(ye^(-3x^2/2))=(-2/3)d(e^(-3x^2/2))
==>∫d(ye^(-3x^2/2))=(-2/3)∫d(e^(-3x^2/2))
==>ye^(-3x^2/2)=(-2/3)e^(-3x^2/2)+C (C是积分常数)
==>y=Ce^(3x^2/2)-2/3
∴此方程的通解是y=Ce^(3x^2/2)-2/3。
(8)解:∵xy'-y=2
==>(xdy-ydx)/x^2=2dx/x^2 (等式两端同乘dx/x^2)
==>d(y/x)=2dx/x^2
==>∫d(y/x)=2∫dx/x^2
==>y/x=C-2/x (C是积分常数)
==>y=Cx-2
∴此方程的通解是y=Cx-2
∵y(1)=0
∴代入通解得 C=2
故所求特解是y=2(x-1)。
说明:此两题也可以直接应用一阶线性微分方程通解公式求解。
(4)解:∵y'-3xy=2x
==>e^(-3x^2/2)dy-3xye^(-3x^2/2)dx=2xe^(-3x^2/2)dx (等式两端同乘e^(-3x^2/2)dx)
==>d(ye^(-3x^2/2))=(-2/3)d(e^(-3x^2/2))
==>∫d(ye^(-3x^2/2))=(-2/3)∫d(e^(-3x^2/2))
==>ye^(-3x^2/2)=(-2/3)e^(-3x^2/2)+C (C是积分常数)
==>y=Ce^(3x^2/2)-2/3
∴此方程的通解是y=Ce^(3x^2/2)-2/3。
(8)解:∵xy'-y=2
==>(xdy-ydx)/x^2=2dx/x^2 (等式两端同乘dx/x^2)
==>d(y/x)=2dx/x^2
==>∫d(y/x)=2∫dx/x^2
==>y/x=C-2/x (C是积分常数)
==>y=Cx-2
∴此方程的通解是y=Cx-2
∵y(1)=0
∴代入通解得 C=2
故所求特解是y=2(x-1)。
说明:此两题也可以直接应用一阶线性微分方程通解公式求解。