7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.
u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³

Verified step by step guidance

1. Identify the given function and the differential equation: the function is \(u(t) = C_1 t^5 + C_2 t^{-4} - t^3\), and the differential equation is \(t^2 u''(t) - 20 u(t) = 14 t^3\).
2. Compute the first derivative \(u'(t)\) by differentiating each term of \(u(t)\) with respect to \(t\): use the power rule for derivatives.
3. Compute the second derivative \(u''(t)\) by differentiating \(u'(t)\) again with respect to \(t\).
4. Substitute \(u(t)\) and \(u''(t)\) into the left-hand side of the differential equation: calculate \(t^2 u''(t) - 20 u(t)\).
5. Simplify the expression obtained after substitution and verify if it equals the right-hand side \$14 t^3\( for all \)t\(, confirming that \)u(t)$ is a solution.

Key Concepts

General Solution of Differential Equations

A general solution includes arbitrary constants representing all possible solutions to a differential equation. Verifying a solution involves substituting it into the equation to check if it satisfies the relationship for all values in the domain.

Second Derivative and Differentiation

The second derivative measures the rate of change of the first derivative and is essential in second-order differential equations. Calculating u''(t) accurately is crucial for substituting into the given differential equation.

Substitution and Simplification in Differential Equations

To verify a solution, substitute the function and its derivatives into the differential equation. Simplify the resulting expression to confirm if both sides of the equation are equal, ensuring the function satisfies the equation.