# -1.3.9 Verify ty'-y=t^2 is a solution of the DE

• MHB
• karush
In summary, a differential equation (DE) is an equation that relates an unknown function to its derivatives and is commonly used in mathematics and science. To verify a solution for a DE, one must substitute the solution into the original equation and check if the equation holds true. The notation "ty'-y=t^2" represents a first-order linear DE and its solution is a function y(t) that satisfies the equation for all values of t. Verifying a solution for a DE is important because it ensures accuracy and can help gain a better understanding of the DE and its properties.
karush
Gold Member
MHB
$\textsf{ Verify the following given functions is a solution of the differential equation}\\ \\$
$ty'-y=t^2\\$
$y_1(t)=3t+t^2$
\begin{align*}
t(3t+t^2)'-(3t+t^2)&=t^2\\
t(3+2t)-(3t+t^2)&=\\
3t+2t^2-3t-t^2&=\\
t^2&=t^2
\end{align*}

probably too easy

Last edited:
Well, learning Calculus does make some problems easy!

## 1. What is a differential equation (DE)?

A differential equation is an equation that relates an unknown function to its derivatives. It is commonly used in mathematics and science to describe how quantities change over time.

## 2. How do you verify a solution for a DE?

To verify a solution for a DE, you substitute the solution into the original equation and check if the equation holds true. In this case, we would substitute -1.3.9 for y and t^2 for t in the equation ty' = t^2 and see if it is a true statement.

## 3. What does "ty'-y=t^2" mean?

This notation represents a first-order linear DE, where y' is the first derivative of y with respect to t. The equation ty'-y=t^2 means that the derivative of y with respect to t multiplied by t is equal to t^2.

## 4. What does it mean for "ty'-y=t^2" to have a solution?

Having a solution for "ty'-y=t^2" means that there exists a function y(t) that satisfies the equation. In other words, when we substitute this function into the equation, it holds true for all values of t.

## 5. Why is it important to verify a solution for a DE?

Verifying a solution for a DE is important because it ensures that the solution is valid and accurate. It also allows us to check if there are any errors in our calculations or if there are any other solutions that we may have missed. Additionally, verifying a solution can help us gain a better understanding of the DE and its properties.

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