Questions: Find the general solution to the homogeneous differential equation. dy^2/dt^2 - 4 dy/dt + 85 y=0 Use c1 and c2 in your answer to denote arbitrary constants, and enter them as c1 and c2. y(t)=c1 e^(-2 t) cos(9 t) + c2 e^(-2 t) sin(9 t)

Solution Steps

To solve the given second-order homogeneous differential equation, we first find the characteristic equation by assuming a solution of the form \( y = e^{rt} \). This leads to a quadratic equation in terms of \( r \). Solving this quadratic equation will give us the roots, which can be real or complex. If the roots are complex, the general solution will involve exponential functions multiplied by sine and cosine functions.

To solve the differential equation

\[ \frac{d^{2} y}{d t^{2}} - 4 \frac{d y}{d t} + 85 y = 0, \]

we assume a solution of the form \( y = e^{rt} \). Substituting this into the differential equation gives the characteristic equation:

\[ r^2 - 4r + 85 = 0. \]

The characteristic equation is a quadratic equation in \( r \). Solving for \( r \) using the quadratic formula:

\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]

where \( a = 1 \), \( b = -4 \), and \( c = 85 \), we find:

\[ r = \frac{4 \pm \sqrt{16 - 340}}{2} = \frac{4 \pm \sqrt{-324}}{2}. \]

This simplifies to:

\[ r = 2 \pm 9i. \]

Since the roots are complex, the general solution to the differential equation is given by:

\[ y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t)), \]

where \( \alpha = 2 \) and \( \beta = 9 \). Thus, the general solution is:

\[ y(t) = e^{2t} (C_1 \cos(9t) + C_2 \sin(9t)). \]

Final Answer