# Is the given function a solution of the differential equation?

• MHB
• karush
In summary, the given functions $y_1(t)=t/3$ and $y_2(t)=e^{-t}+t/3$ are verified to be solutions of the differential equation $y''''+4y'''+3y=t$.
karush
Gold Member
MHB
$\textsf{ Verify the following given functions is a solution of the differential equation}\\ \\$
$y''''+4y'''+3y=t\\$
$y_1(t)=t/3$
\begin{align*}
(t/3)''''+4(t/3)'''+(t/3)&=t\\
0+0+t&=t
\end{align*}
$y_2(t)=e^{-t}+t/3$
\begin{align*}
(e^{-t}+t/3)''''+4(e^{-t}+t/3)'''+3(e^{-t}+t/3)&=t\\
e^{-t}-4e^{-t}+3e^{-t}+3e^{-t}+t&=t\\
t&=t
\end{align*}

so is it Raj now?

$$\tiny{\textsf{Elementary Differential Equations and Boundary Value Problems}}$$

A bunch of typos but it's correct. First one it's $3 \cdot (t/3)$, and second one you copied $3e^{-t}$ twice.

$\textsf{ Verify the following given functions is a solution of the differential equation}\\ \\$
$y''''+4y'''+3y=t\\$
$y_1(t)=t/3$
\begin{align*}
(t/3)''''+4(t/3)'''+3(t/3)&=t\\
0+0+t&=t
\end{align*}
$y_2(t)=e^{-t}+t/3$
\begin{align*}
(e^{-t}+t/3)''''+4(e^{-t}+t/3)'''+3(e^{-t}+t/3)&=t\\
e^{-t}-4e^{-t}+3e^{-t}+t&=t\\
t&=t
\end{align*}

Rido12 said:
A bunch of typos but it's correct. First one it's $3 \cdot (t/3)$, and second one you copied $3e^{-t}$ twice.
mahalo

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It describes the relationship between the rate of change of a function and the function itself.

## 2. What is the process of verifying a function as a solution of a differential equation?

To verify a function as a solution of a differential equation, you must plug the function into the equation and check that it satisfies the equation for all values of the independent variable. This means that when you take the derivative of the function and substitute it into the equation, the resulting expression should equal the original function.

## 3. Why is it important to verify a function as a solution of a differential equation?

Verifying a function as a solution of a differential equation is important because it ensures that the function accurately represents the relationship between the rate of change of a system and the system itself. It also helps to check for any errors in the solution and ensure its validity.

## 4. Can a function be a solution of a differential equation without being verified?

No, a function cannot be considered a solution of a differential equation without being verified. Verification is necessary to ensure that the function accurately represents the relationship described by the equation and to check for any errors in the solution.

## 5. What are the common methods used for verifying a function as a solution of a differential equation?

The most common methods for verifying a function as a solution of a differential equation include substitution, separation of variables, and variation of parameters. These methods involve manipulating the equation and the function to check for equality and determine the validity of the solution.

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