y-y-x-2x-3-cos-x-2-y-where-y-pi-0-

Question Number 116136 by bemath last updated on 01/Oct/20

y^{'} = \frac{y}{x} + \frac{{2 x}^{3} \cos (x^{2})}{y}

where y (\sqrt{π}) = 0

Answered by mindispower last updated on 01/Oct/20

\frac{y}{x} = z

y^{'} = z + {x z}^{'}

\Rightarrow z + {x z}^{'} = z + \frac{2 x^{3} c o s (x^{2})}{x z}

\Rightarrow z ({x z}^{'}) = 2 x^{2} c o s (x^{2})

\Rightarrow {z z}^{'} = 2 x c o s (x^{2})

\Rightarrow \int z d z = \int 2 x c o s (x^{2}) d x

\Leftrightarrow \frac{z^{2}}{2} = s i n (x^{2}) + c

z = \underline{+} 2 \sqrt{s i n (x^{2}) + c}

y = \underline{+} z \sqrt{s i n (x^{2}) + c}

y (\sqrt{π}) = \underline{+} 2 \sqrt{π} . \sqrt{s i n (π) + c} = 0 \Rightarrow c = 0

y (x) = \underline{+} 2 \sqrt{s i n (x^{2})}

Answered by TANMAY PANACEA last updated on 01/Oct/20

\frac{d y}{d x} - \frac{y}{x} = \frac{2 x^{3} {c o s x}^{2}}{y}

\frac{x d y - y d x}{x d x \times x^{2}} = \frac{2 {c o s x}^{2}}{(\frac{y}{x})}

d (\frac{y}{x}) . \frac{y}{x} = {c o s x}^{2} . {d x}^{2}

i n t r e g a t i n g

\int (\frac{y}{x}) d (\frac{y}{x}) = \int {c o s x}^{2} . {d x}^{2}

\frac{1}{2} {(\frac{y}{x})}^{2} = {s i n x}^{2} + c

Answered by mathmax by abdo last updated on 01/Oct/20

y^{^{'}} = \frac{y}{x} + \frac{{2 x}^{3} \cos (x^{2})}{y} let \frac{y}{x} = z \Rightarrow y = xz \Rightarrow y^{^{'}} = z + {xz}^{^{'}}

(e) \Rightarrow z + {xz}^{^{'}} = z + \frac{{2 x}^{2} \cos (x^{2})}{z} \Rightarrow {xzz}^{^{'}} = {2 x}^{2} \cos (x^{2}) \Rightarrow

{zz}^{^{'}} = 2 xcos (x^{2}) \Rightarrow \int {zz}^{^{'}} dx = 2 \int xcos (x^{2}) dx = \sin (x^{2}) + c \Rightarrow

\frac{1}{2} z^{2} = \sin (x^{2}) + c \Rightarrow z^{2} = 2 \sin (x^{2}) + 2 c \Rightarrow z = \bar{+} \sqrt{2 \sin (x^{2}) + λ}

Commented by mathmax by abdo last updated on 01/Oct/20

\Rightarrow y = \bar{+} x \sqrt{2 \sin (x^{2}) + λ}

y (\sqrt{π}) = 0 \Rightarrow \bar{+} \sqrt{π} \sqrt{0 + λ} = 0 \Rightarrow λ = 0 \Rightarrow y = \bar{+} \sqrt{2 \sin (x^{2})}

Commented by mathmax by abdo last updated on 02/Oct/20

y = \bar{+} x \sqrt{2 \sin (x^{2})}