x2+a2dydx+y=x2+a2-x =dydx+1x2+a2y= 1-xx2+a2The integrating factor u= e∫dxx2+a2=x2+a2+xy=1u(x)∫u(x) 1-xx2+a2dx = 1x2+a2+x∫x2+a2+x 1-xx2+a2dx = a2ln( a2+x2+x)x2+a2+x

solve

\sqrt{x^{2} + a^{2}} \frac{d y}{d x} + y = \sqrt{x^{2} + a^{2}} - x = \frac{d y}{d x} + \frac{1}{\sqrt{x^{2} + a^{2}}} y = 1 - \frac{x}{\sqrt{x^{2} + a^{2}}} T h e i n t e g r a t i n g f a c t o r u = e^{\int_{}^{} \frac{d x}{\sqrt{x^{2} + a^{2}}}} = (\sqrt{x^{2} + a^{2}} + x) y = \frac{1}{u (x)} \int_{}^{} u (x) (1 - \frac{x}{\sqrt{x^{2} + a^{2}}}) d x = \frac{1}{(\sqrt{x^{2} + a^{2}} + x)} \int_{}^{} (\sqrt{x^{2} + a^{2}} + x) (1 - \frac{x}{\sqrt{x^{2} + a^{2}}}) d x = \frac{a^{2} \ln (\sqrt{a^{2} + x^{2}} + x)}{(\sqrt{x^{2} + a^{2}} + x)}

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