arithmetic sequence; two differential equations

[Ozi]

I need help solving these

Question 3: Let a₁, a₂, a₃, ... .., a₁, ... be an arithmetic sequence such that a₃ = 8 and a₅ = 15.

Find the difference d = a_n+1 - a_n for n = 1, 2, ...

Question 4: (a) Find the particular solution of

. . . . .\(\displaystyle \dfrac{dy}{dx}\, =\, \dfrac{y^2\, -\, 1}{x}\, \mbox{ with }\, y(1)\, =\, 2\)

(b) Find the general solution of the equation \(\displaystyle y"\, -\, 6y'\, +\, 25y\, =\, 0\)

thanks

 

[stapel]

Question 3: Let a₁, a₂, a₃, ... .., a₁, ... be an arithmetic sequence such that a₃ = 8 and a₅ = 15.

Find the difference d = a_n+1 - a_n for n = 1, 2, ...

This is a simple algebra exercise. How far did you get in applying the formulas for sequences (here) and series (here)? You noted that the fifth term was the third term plus twice the common difference. What then is the common difference? What did you do next?

Question 4: (a) Find the particular solution of

. . . . .\(\displaystyle \dfrac{dy}{dx}\, =\, \dfrac{y^2\, -\, 1}{x}\, \mbox{ with }\, y(1)\, =\, 2\)

This is a simple differential equation; just separate and integrate!

(b) Find the general solution of the equation \(\displaystyle y"\, -\, 6y'\, +\, 25y\, =\, 0\)

What did you attempt here?

When you reply, please show all of your efforts so far. Thank you!