Questions: Find the general solution of the differential equation 121 y'' + 176 y' + 64 y = 0 . Use c1, c2 for the constants of integration. Use t for the independent variable. y(t)=

Solution Steps

To solve the given second-order linear homogeneous differential equation with constant coefficients, we first find the characteristic equation. The characteristic equation is obtained by replacing \( y'' \) with \( r^2 \), \( y' \) with \( r \), and \( y \) with 1, resulting in a quadratic equation. We then solve this quadratic equation to find the roots. Depending on the nature of the roots (real and distinct, real and repeated, or complex), we write the general solution using the appropriate form.

The given differential equation is

\[ 121 y^{\prime \prime} + 176 y^{\prime} + 64 y = 0. \]

To find the general solution, we first form the characteristic equation by substituting \( y'' \) with \( r^2 \), \( y' \) with \( r \), and \( y \) with 1:

\[ 121 r^2 + 176 r + 64 = 0. \]

We solve the characteristic equation for \( r \). The roots obtained are:

\[ r = -\frac{8}{11}. \]

Since we have a repeated root, we can denote it as \( r_1 = r_2 = -\frac{8}{11} \).

For a second-order linear differential equation with a repeated root \( r \), the general solution is given by:

\[ y(t) = (c_1 + c_2 t) e^{r t}. \]

Substituting the value of \( r \):

\[ y(t) = \left(c_1 + c_2 t\right) e^{-\frac{8}{11} t}. \]

Final Answer