(1+y^2)dx=(y-sqrt(1+y^2))(1+x^2)^{3/2}dy

解答

(1 + y^{2}) d x = (y - 1 + y^{2}) (1 + x^{2})^{\frac{3}{2}} d y

解答

y = \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} + 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = - \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} + 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} - 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = - \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} - 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}

求解步骤

(1 + y^{2}) d x = (y - 1 + y^{2}) (1 + x^{2})^{\frac{3}{2}} d y

求解可分离方程:

y = \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} + 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = - \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} + 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} - 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = - \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} - 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}

y = \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} + 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = - \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} + 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} - 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}, y = - \frac{3 e ^{2 c_{1} + \frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - e ^{4 c_{1}} - 9 e ^{\frac{4 ( x + c _{1} ( x ^{2} + 1 ) ^{\frac{1}{2}} )}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + 4 e ^{2 c_{1} + \frac{6 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} - 9 e ^{4 c_{1} + \frac{4 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} ^{\frac{1}{2}}}{e ^{2 c_{1}} ( - 4 e ^{\frac{2 x}{( x ^{2} + 1 ) ^{\frac{1}{2}}}} + e ^{2 c_{1}} )}

作图

流行的例子

(x^2+1)dy+xy^2dx=0

(x^{2} + 1) d y + x y^{2} d x = 0

x+y(dy)/(dx)=0

x + y \frac{d y}{d x} = 0

y^'=(1+y^2)cos(t)

y^{^{'}} = (1 + y^{2}) cos (t)

xy^2y^'=x+10

x y^{2} y^{^{'}} = x + 10

(dy}{dx}=\frac{3e^x)/y

\frac{d y}{d x} = \frac{3 e ^{x}}{y}