Find the complete general solution to the 2nd ODE:
9y'' + 9y' - 4y = 0
9a^2+9a-4=0
(3a-1)(3a+4) = 0
a = -4/3,1/3
y = c1 e^(-4/3 x) + c2 e^(1/3 x)
Thank you so much Steve! I had gotten this far but what about solving for the "complete" general solution? That's where I get lost....
you got me.
We have found the general solution. Dunno what more you can come up with for a "complete" general solution.
Does your text explain the difference? Sounds bogus to me.
To find the complete general solution to the given second-order ordinary differential equation (ODE), we can follow these steps:
Step 1: Write the characteristic equation.
The characteristic equation for the given ODE is obtained by substituting y = e^(rt) into the equation, where r is an unknown constant:
9r^2 + 9r - 4 = 0.
Step 2: Solve the characteristic equation.
By factoring or using the quadratic formula, we can solve the characteristic equation as follows:
(3r - 1)(3r + 4) = 0.
This gives two distinct roots: r1 = 1/3 and r2 = -4/3.
Step 3: Determine the general solution.
The general solution is expressed as a linear combination of exponentials, using the roots obtained in Step 2:
y(t) = C1e^(r1t) + C2e^(r2t),
where C1 and C2 are arbitrary constants to be determined.
Step 4: Substitute the roots back into the general solution and simplify.
By plugging in the roots r1 = 1/3 and r2 = -4/3 into the general solution, and simplifying, we get:
y(t) = C1e^(t/3) + C2e^(-4t/3).
Therefore, the complete general solution to the given second-order ODE is:
y(t) = C1e^(t/3) + C2e^(-4t/3),
where C1 and C2 are arbitrary constants.