Final answer:
A particular solution to the differential equation can be found using the method of undetermined coefficients or variation of parameters after correcting the typo in the given equation.
Step-by-step explanation:
To find a particular solution to the differential equation y'+6 y'+5 y=10 t e³⁴, we need to use an appropriate method for nonhomogeneous equations, such as the method of undetermined coefficients or variation of parameters. However, there seems to be a typo in the provided equation, as y' is written twice. Assuming the correct equation is y''+6 y'+5 y=10 t e³⁴, we would seek a solution of the form Ate³⁴, where A is a constant to be determined. To find the value of A, we would differentiate our proposed form twice, plug the results into the original equation, and compare coefficients to solve for A