Integrating 1/(y²-1) for separable ODE

[The-Mad-Lisper]

Homework Statement

$$\frac{dy}{dx}=y^2-1$$
$$y(0)=3$$

Homework Equations

$$\frac{dy}{dx}=f(y) \leftrightarrow \frac{dx}{dy}=\frac{1}{f(y)}$$

The Attempt at a Solution

$$\frac{dx}{dy}=\frac{1}{y^2-1}$$
$$dx=\frac{dy}{y^2-1}$$
$$\int dx=\int \frac{dy}{y^2-1}+C$$
$$x=\int \frac{dy}{y^2-1}+C$$
How do I integrate $\int \frac{dy}{y^2-1}$?

 

[Mark44]

Homework Statement

$$\frac{dy}{dx}=y^2-1$$
$$y(0)=3$$

Homework Equations

$$\frac{dy}{dx}=f(y) \leftrightarrow \frac{dx}{dy}=\frac{1}{f(y)}$$

The Attempt at a Solution

$$\frac{dx}{dy}=\frac{1}{y^2-1}$$
$$dx=\frac{dy}{y^2-1}$$
$$\int dx=\int \frac{dy}{y^2-1}+C$$
$$x=\int \frac{dy}{y^2-1}+C$$
How do I integrate $\int \frac{dy}{y^2-1}$?

Partial fractions. See https://www.physicsforums.com/insights/partial-fractions-decomposition/ if you are uncertain about this technique.

 

[Buzz Bloom]

Hi Mad:

I will give you a hint. think about factoring y²-1 = f₁(y) × f₂(y).
Then think about finding A and B such that 1/(f₁ × f₂) = A/f₁ + B/f₂.

Hope this helps.

Regards,
Buzz

 

[The-Mad-Lisper]

Thanks, I got the answer.

 

[epenguin]

Same integral in two ongoing problems if I am not mistaken.

https://www.physicsforums.com/threads/difficult-integral-falling-mass.864659/