Solve-y-y-x-2-

Question Number 14478 by tawa tawa last updated on 01/Jun/17

Solve : y^{'} = {(y - x)}^{2}

Answered by mrW1 last updated on 01/Jun/17

l e t t = y - x

\Rightarrow y = t + x

\frac{d y}{d x} = \frac{d t}{d x} + 1

\Rightarrow \frac{d t}{d x} + 1 = t^{2}

\frac{d t}{t^{2} - 1} = d x

\int \frac{d t}{t^{2} - 1} = \int d x

\frac{1}{2} \int (\frac{1}{t - 1} - \frac{1}{t + 1}) d t = \int d x

\ln \frac{t - 1}{t + 1} = 2 x + C_{1}

\ln \frac{y - x - 1}{y - x + 1} = 2 x + C_{1}

\frac{y - x - 1}{y - x + 1} = {c e}^{2 x}

Commented by tawa tawa last updated on 01/Jun/17

God bless you sir .

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 01/Jun/17

x = r c o s φ, y = r s i n φ

y^{^{'}} = r^{2} {(c o s φ - s i n φ)}^{2} = r^{2} (1 - s i n 2 φ)

y + C = \int r^{2} (1 - s i n 2 φ) d φ = r^{2} φ + \frac{r^{2}}{2} c o s 2 φ

2 y + D = r^{2} (2 φ + c o s 2 φ)

2 y + D = (x^{2} + y^{2}) (2 {t g}^{- 1} (\frac{y}{x}) + \frac{x^{2} - y^{2}}{x^{2} + y^{2}})

Commented by tawa tawa last updated on 01/Jun/17

God bless you sir