91Ó°ÊÓ

When tackling differential equations, verifying a solution is a crucial step. It involves substituting back into the original equation to see if it holds true. For our exercise, we have the function \( y = t^4 \). To verify if it's a solution to the differential equation \( t \frac{d y}{d t} = 4y \), we follow a simple process. First, differentiate \( y \) with respect to \( t \) to find \( \frac{d y}{d t} \). Then, both our original function and its derivative need to be plugged into the differential equation.
By substituting \( \frac{d y}{d t} = 4t^3 \) and \( y = t^4 \) into the equation, we form \( t(4t^3) = 4(t^4) \).
Finally, simplify both sides of this equation. If they match, as they do in this case \( 4t^4 = 4t^4 \), our initial function is verified as a correct solution. This process ensures that the function satisfies the differential equation under the given conditions.

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which gives you the rate of change. It’s like figuring out how a variable quantity changes at any given point. In our problem, we started with \( y = t^4 \).
The differentiation process required us to compute \( \frac{d y}{d t} \), which gives the slope of the function \( y \) or how \( y \) changes concerning \( t \). This involves using differentiation rules such as the power rule. By applying these rules, we discover that \( \frac{d y}{d t} = 4t^3 \).
Differentiating functions correctly is essential when solving differential equations. This allows solving problems where functions may not appear in explicit form, such as motion problems, economic models, and more.

The power rule is a quick and efficient way of differentiating functions where variables are raised to a power. It’s a handy tool in calculus that simplifies the process of finding derivatives. In the problem at hand, we used the power rule to differentiate \( y = t^4 \).
The rule states that for any function \( x^n \), its derivative is \( nx^{n-1} \). Therefore, applying the power rule to \( t^4 \) results in \( 4t^3 \).
Employing the power rule makes it easy to handle polynomial functions with any exponent. This technique is particularly useful because it saves time. It also provides accuracy in obtaining derivatives, which can then be used to check or predict real-world scenarios through differential equations.