The interval notation in which solutions are certain to exist for the given differential equation y^(4) + 7y^''' + 5y = t is (-∞, ∞).
The given differential equation y^(4) + 7y^''' + 5y = t is a linear non-homogeneous equation where y is a dependent variable and t is an independent variable. Here, the degree of the differential equation is 4, and the highest-order derivative is y^4. Since the degree is 4, the given differential equation will have four roots. Therefore, the interval notation in which solutions are certain to exist for the given differential equation y^(4) + 7y^''' + 5y = t is (-∞, ∞). Here, the interval notation is open since there are no restrictions on the independent variable t, and it can take any value. Hence the answer is (-∞, ∞).
Another approach to determine the intervals in which solutions are sure to exist for the given equation y^(4) + 7y^''' + 5y = t, is to use the Existence and Uniqueness Theorem for differential equations. According to this theorem, if the coefficients of the differential equation are continuous functions, then a unique solution exists for any initial condition in the interval. In this case, the coefficients are constants: 1 for y^(4), 7 for y^''', and 5 for y. Since constants are continuous functions, a unique solution exists for any initial condition. Therefore, the interval in which solutions are sure to exist is the entire real line, which in interval notation is written as (-∞, ∞).