solve dy/dx=x*sqrt(x^2-1)/y — step-by-step solution

1) Solve the separable equation

y' = x \cdot \frac{\sqrt{x^{2} - 1}}{y}

, where

y

is a function of

x

and

y

≠ 0

⇥1.1) Let's evaluate the domain of

x \cdot \sqrt{x^{2} - 1}

⇥ For the expression

\sqrt{x^{2} - 1}

to exist, you must check:

x^{2} - 1 >= 0

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

⇥

Δ = 0^{2} - 4 \cdot 1 \cdot (-1)

⇥

Δ = - 4 \cdot (-1)

⇥

Δ = 4

⇥Since

Δ > 0

, the equation has two solutions:

⇥For the first solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

⇥

x1 = - \frac{\sqrt{4}}{2}

⇥

x1 = - \frac{2}{2}

⇥

x1 = -1

⇥Solution 2nd:

\frac{- b + \sqrt{Δ}}{2 a}

⇥

x2 = \frac{\sqrt{4}}{2}

⇥

x2 = \frac{2}{2}

⇥

x2 = 1

⇥Solution found:

x \in (-∞, -1] \cup [1, +∞)

Multiply by

y

and separate the variables:

y

y

x \cdot \sqrt{x^{2} - 1}

x

⇥1.2) Let's compute the integral:

\int(x \cdot \sqrt{x^{2} - 1})dx

⇥For the math expression:

x \cdot \sqrt{x^{2} - 1}

, we know the derivative formula:

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, so we try differentiating with ^(m+1)

⇥1.3) Let's differentiate:

\frac{d}{d x} ({(x^{2} - 1)}^{(\frac{1}{2} + 1)})

⇥

\frac{d}{d x} ({(x^{2} - 1)}^{(\frac{1 + 2}{2})})

⇥

\frac{d}{d x} ({(x^{2} - 1)}^{\frac{3}{2}})

⇥Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

with f =

x^{2} - 1

⇥

\frac{3}{2} \cdot \frac{d}{d x} (x^{2} - 1) \cdot {(x^{2} - 1)}^{(\frac{3}{2} - 1)}

⇥ℹ

\frac{3}{2} \cdot \frac{d}{d x} (x^{2} - 1) \cdot {(x^{2} - 1)}^{(\frac{3}{2} - 1)}

\frac{3 \cdot \frac{d}{d x} (x^{2} - 1) \cdot {(x^{2} - 1)}^{(\frac{3}{2} - 1)}}{2}

‖

\frac{d}{d x} (x^{2} - 1)

\frac{d}{d x} (x^{2}) + 0

‖

\frac{3}{2} - 1

\frac{3 - 1 \cdot 2}{2}

⇥

\frac{3 \cdot (\frac{d}{d x} (x^{2}) + 0) \cdot {(x^{2} - 1)}^{(\frac{3 - 1 \cdot 2}{2})}}{2}

⇥ℹ

\frac{d}{d x} (x^{2})

2 x

‖

-1 \cdot 2

-2

⇥

\frac{3 \cdot 2 x \cdot {(x^{2} - 1)}^{(\frac{3 - 2}{2})}}{2}

⇥ℹ

3 \cdot 2

6

‖

3 - 2

1

⇥

\frac{6 x \cdot \sqrt{x^{2} - 1}}{2}

⇥

3 x \cdot \sqrt{x^{2} - 1}

⇥You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

⇥

\frac{1}{3} \cdot {(x^{2} - 1)}^{\frac{3}{2}}

⇥

\frac{{(x^{2} - 1)}^{\frac{3}{2}}}{3}

Integrating both sides gives

\frac{y^{2}}{2} = \frac{{(x^{2} - 1)}^{\frac{3}{2}}}{3} + C

, where

C

is an arbitrary real constant

Multiply by 2 and rename 2*

C

C

y^{2} = \frac{2 \cdot {(x^{2} - 1)}^{\frac{3}{2}}}{3} + C

The two real branches have a fixed sign on each interval where the original right side is real and

\frac{2 \cdot {(x^{2} - 1)}^{\frac{3}{2}}}{3} + C

> 0;

y

= 0 is excluded

▶Solution:

y = {- \sqrt{\frac{2 \cdot {(x^{2} - 1)}^{\frac{3}{2}}}{3} + C}, \sqrt{\frac{2 \cdot {(x^{2} - 1)}^{\frac{3}{2}}}{3} + C}}