How do you solve the differential equation $\frac{d y}{d x} = 7 x \sqrt{y}$ $dy/dx = 7x sqrt(y)$ ?

Question

Solve the differential equation.

$\frac{d y}{d x} = 7 x \sqrt{y}$ $dy/dx = 7x sqrt(y)$ for $y \neq 0$ $y != 0$

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Answer 1 · 2017-06-28T10:28:42

$y = \frac{1}{16} {(7 x^{2} + A)}^{2}$ $y= 1/16(7x^2+A)^2$

We have:

$\frac{d y}{d x} = 7 x \sqrt{y}$ $dy/dx = 7xsqrt(y)$

Which if we collect terms, (as $y \neq 0$ $y ne0$ ) can be written as:

$1/sqrt(y) \ dy/dx = 7x$

Which is a First Order non-linear separable Differential Equation, so we can "separate the variables" to get:

$int \ 1/sqrt(y) \ dy = int \ 7x \ dx$

Which we can integrate to get:

$2sqrt(y)= (7x^2)/2 + C$

And now we can form an explicit solution:

$sqrt(y)= (7x^2)/4 + C/2$

$:. sqrt(y)= 1/4(7x^2+A)$
$:. y= 1/16(7x^2+A)^2$

Which, is the General Solution.