(1-y^2)^{0.5}dx=(1+x^2)ydy

해법

(1 - y^{2})^{0.5} d x = (1 + x^{2}) y d y

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

솔루션 단계

(1 - y^{2})^{0.5} d x = (1 + x^{2}) y d y

분리 가능한 ODE 해결:

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

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