Resolve-x-2-y-xy-y-1-

Question Number 168722 by LEKOUMA last updated on 16/Apr/22

R e s o l v e

x^{2} y^{^{″}} + {x y}^{^{'}} + y = 1

Commented by mokys last updated on 16/Apr/22

s o l u t i o n

l e t : z = l n x \to x = e^{z}

[D (D - 1) + D + 1] y = 1

(D^{2} + 1) y = 1

(m^{2} + 1) = 0 \Rightarrow m = \pm i

y_{c} = C_{1} c o s (z) + C_{2} s i n (z)

y_{c} = C_{1} c o s (l n x) + C_{2} s i n (l n x)

l e t : y_{p} = a \to y_{p}^{^{'}} = y_{p}^{^{″}} = 0

∴ 0 + a = 1 \Rightarrow a = 1

∴ y_{p} = 1

y = y_{c} + y_{p} = C_{1} c o s (l n x) + C_{2} s i n (l n x) + 1

((a l d o l a i m y))

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