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dy/dx+xsin2y=x³cos²y

\frac{d y}{d x} + x \sin 2 y = x^{3} \cos^{2} y d i v i d i n g b o t h s i d e s b y \cos^{2} y, w e g e t \frac{1}{\cos^{2} y} \frac{d y}{d x} + \frac{x \sin 2 y}{\cos^{2} y} = x^{3} s e c^{2} y \frac{d y}{d x} + 2 x \tan x = x^{3} L e t \tan x = t, s e c^{2} y \frac{d y}{d x} = \frac{d t}{d x} H e n c e \frac{d t}{d x} + 2 x t = x^{3} t h i s i s l i n e a r f o r m, s o i n t e g r a t i n g f a c t o r w i l l b e e^{\int 2 x d x} = e^{x^{2}} S o t \times e^{x^{2}} = \int x^{3} \times e^{x^{2}} d x L e t e^{x^{2}} = u, t h e n 2 x e^{x^{2}} d x = d u, a n d \log u = x^{2} H e n c e t \times e^{x^{2}} = \frac{1}{2} \int \log u d u O r t \times e^{x^{2}} = \frac{1}{2} (u \log u - u) = \frac{1}{2} (x^{2} e^{x^{2}} - e^{x^{2}}) O r \tan x \times e^{x^{2}} = \frac{1}{2} (x^{2} e^{x^{2}} - e^{x^{2}}) + C

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