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

해법

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

해법

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}} )}

솔루션 단계

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

분리 가능한 ODE 해결:

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}} )}

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