求下列微分方程的通解,xdy/dx=(yIn^2)y,[(y+1)^2]dy/dx+x^3=0,dy/dx=2^(x+y),帮忙算下,给过程 6x+y
求下列微分方程的通解,xdy/dx=(yIn^2)y,[(y+1)^2]dy/dx+x^3=0,dy/dx=2^(x+y),帮忙算下,帮忙给过程,谢谢,我有急用...
求下列微分方程的通解,xdy/dx=(yIn^2)y,[(y+1)^2]dy/dx+x^3=0,dy/dx=2^(x+y),帮忙算下,帮忙给过程,谢谢,我有急用
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1.求xdy/dx=yIn²y通解
解:∵xdy/dx=yIn²y ==>dy/(yIn²y)=dx/x
==>d(lny)/In²y=dx/x
==>-1/lny=ln│x│+C (C是积分常数)
经检验y=1也是原方程的解
∴原方程的通解是y=1或-1/lny=ln│x│+C (C是积分常数);
2.求[(y+1)²]dy/dx+x³=0通解
解:∵[(y+1)²]dy/dx+x³=0 ==>[(y+1)²]dy=-x³dx
==>(y+1)³/3=C/3-x^4/4 (C是积分常数)
==>(y+1)³=C-3x^4/4
∴原方程的通解是(y+1)³=C-3x^4/4 (C是积分常数);
3.求dy/dx=2^(x+y)通解
解:∵dy/dx=2^(x+y) ==>dy/dx=(2^x)(2^y)
==>dy/2^y=2^xdx
==>e^(-yln2)dy=e^(xln2)dx
==>e^(-yln2)d(-yln2)=-e^(xln2)d(xln2)
==>e^(-yln2)=C-e^(xln2) (C是积分常数)
==>2^(-y)=C-2^x
∴原方程的通解是2^(-y)=C-2^x (C是积分常数)。
解:∵xdy/dx=yIn²y ==>dy/(yIn²y)=dx/x
==>d(lny)/In²y=dx/x
==>-1/lny=ln│x│+C (C是积分常数)
经检验y=1也是原方程的解
∴原方程的通解是y=1或-1/lny=ln│x│+C (C是积分常数);
2.求[(y+1)²]dy/dx+x³=0通解
解:∵[(y+1)²]dy/dx+x³=0 ==>[(y+1)²]dy=-x³dx
==>(y+1)³/3=C/3-x^4/4 (C是积分常数)
==>(y+1)³=C-3x^4/4
∴原方程的通解是(y+1)³=C-3x^4/4 (C是积分常数);
3.求dy/dx=2^(x+y)通解
解:∵dy/dx=2^(x+y) ==>dy/dx=(2^x)(2^y)
==>dy/2^y=2^xdx
==>e^(-yln2)dy=e^(xln2)dx
==>e^(-yln2)d(-yln2)=-e^(xln2)d(xln2)
==>e^(-yln2)=C-e^(xln2) (C是积分常数)
==>2^(-y)=C-2^x
∴原方程的通解是2^(-y)=C-2^x (C是积分常数)。
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都是变量分离方程。
xdy/dx=(yIn^2)y:变量分离得:dy/(yIn^2)y=dx/x,两边积分得通解为:-1/lny=ln|x|+C
[(y+1)^2]dy/dx+x^3=0:变量分离得:[(y+1)^2]dy=-x^3dx,
两边积分得通解为:(y+1)^3*1/3=x^4/4+C
dy/dx=2^(x+y):变量分离得:1/2^(y)*dy=2^(x)dx,
两边积分得通解为:-1/(2^y*ln2)=2^x/ln2+C
xdy/dx=(yIn^2)y:变量分离得:dy/(yIn^2)y=dx/x,两边积分得通解为:-1/lny=ln|x|+C
[(y+1)^2]dy/dx+x^3=0:变量分离得:[(y+1)^2]dy=-x^3dx,
两边积分得通解为:(y+1)^3*1/3=x^4/4+C
dy/dx=2^(x+y):变量分离得:1/2^(y)*dy=2^(x)dx,
两边积分得通解为:-1/(2^y*ln2)=2^x/ln2+C
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dy/dx=根号下(1-y^2)/(1-x^2),cosydx-[1+e^(-x)]sinydy=0,再帮忙算下,谢谢
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呵呵,是一个人哟,已经算了,去看看
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