Need Help with Integration for Solving 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/