Question: "Solve for G.S / P.S forbthe following differential..."

High School
Algebra
Algebra I
Forms of linear equations

Khadija Wells

Answered question

2021-01-22

Solve for G.S / P.S forbthe following differential equations using the method of solutions for linear equations of first order, first degree

(y - x) (y + x) (y^{2} + x^{2}) (x d y - y d x) = x^{6} d x

Answer & Explanation

Macsen Nixon

Skilled2021-01-23Added 117 answers

$(y - x) (y + x) (y^{2} + x^{2}) (x d y - y d x) = x^{6} d x$
$\Rightarrow (y^{2} - x^{2}) (y^{2} - x^{2}) (x d y - y d x) = x^{6} d x$
$\Rightarrow (y^{4} - x^{4}) (x d y - y d x) = x^{4} d x [(a + b) (a - b) = a^{2} - b^{2}]$
$\Rightarrow \frac{(y^{4} - x^{4}} {x^{4}} \frac{x d y - y d x}{x^{2}} = d x}{}$
$[{(\frac{y}{x})}^{4} - 1] d \frac{y}{x} = d x$
$\int [{(\frac{y}{x})}^{4} - 1] d \frac{y}{x} = \int d x + e (e = c o n s \tan t)$
$\Rightarrow \frac{{\frac{y}{x}}^{4 + 1}}{4 + 1} - \frac{y}{x} = x + e$
$\Rightarrow \frac{y^{5}}{5 x^{5}} - \frac{y}{x} = x + e$
$\Rightarrow y^{5} - 5 x 6 {4} y = 5 x^{5} (x + e)$
$\Rightarrow y (y^{4} - 5 x^{4}) = 5 x^{5} (x + e)$
Therefore the sol is
$y (y^{4} - 5 x^{4}) = 5 x^{5} (x + e)$

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