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To determine the interval where solutions are guaranteed to exist for the differential equation:

\[
y^{(4)} + 6y'' + 4y = t
\]

we analyze the equation as a linear, non-homogeneous differential equation with constant coefficients. Since all the coefficients of \( y \), \( y'' \), and \( y^{(4)} \) are constants (6 and 4), this equation is valid for all real values of \( t \). Therefore, the interval of validity where solutions are sure to exist is:

\[
(-\infty, \infty)
\]

Would you like further explanation on how to analyze such intervals or details on solving similar differential equations?

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Here are some related questions to explore:

1. How do we determine the interval of existence for solutions in non-linear differential equations?
2. What is the characteristic equation for a differential equation with constant coefficients?
3. How do homogeneous and non-homogeneous differential equations differ in their solution intervals?
4. What role does the highest derivative play in determining the solution's interval?
5. How can we apply initial conditions to further specify solutions?

**Tip:** In linear differential equations with constant coefficients, if there are no points of singularity, solutions often exist over all real numbers.

Determine the interval in which solutions are sure to exist for the equation y^(4) + 6y'' + 4y = t.

Learn how to determine the interval in which solutions are guaranteed to exist for the differential equation y^(4) + 6y'' + 4y = t. This problem covers linear differential equations with constant coefficients and the Existence and Uniqueness Theorem. Suitable for college-level students.

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