# -b.2.1.5 Find the general solution of y' - 2y =3te^t,

• MHB
• karush
In summary, the conversation was about finding the general solution of a given differential equation and obtaining $u(t)$. The expert summarizer provided the correct solution as $y=-3e^t t - 3e^{-t} +c_1 e^t$, while noting that the solution given by the textbook was incorrect. The importance of being careful in solving these types of problems was also mentioned.
karush
Gold Member
MHB
\nmh{895}
Find the general solution of the given differential equation
$\displaystyle y^\prime - 2y =3te^t, \\$

Obtain $u(t)$
$\displaystyle u(t)=\exp\int -2 \, dx =e^{-2t} \\$
Multiply thru with $e^{-2t}$
$e^{-2t}y^\prime + 2e^{-2t}y= 3te^{-t} \\$
Simplify:
$(e^{-2t}y)'= 3te^{-t} \\$
Integrate:
$\displaystyle e^{-2t}y=\int 3te^{-t} dt =-3e^{-t}(t+1)+c_1 \\$
{Divide by $e^{-2t}$
$y=-3e^t t - 3e^{-t} +c_1 e^t \\$
$y=\color{red}{c_1 e^{2t}-3e^t}$

ok sumtum went wrong somewhere?

$$\tiny\color{blue}{\textbf{Text book: Elementary Differential Equations and Boundary Value Problems}}$$

Last edited:
I with you up to after the integration:

$$\displaystyle e^{-2t}y=-3e^{-t}(t+1)+c_1$$

Thus:

$$\displaystyle y(t)=-3e^{t}(t+1)+c_1e^{2t}$$

The solution given by the book is incorrect.

these problems require mega being carefull!

## 1. What is the general solution of y' - 2y =3te^t?

The general solution of y' - 2y =3te^t is y(t) = C*e^(-2t) + t^2*e^t, where C is a constant.

## 2. How do you find the general solution of y' - 2y =3te^t?

To find the general solution, we first solve for the homogeneous solution by setting the non-homogeneous term (3te^t) to 0. Then, we use the method of undetermined coefficients to find a particular solution. The general solution is the sum of the homogeneous and particular solutions.

## 3. What is the purpose of the constant C in the general solution?

The constant C represents the arbitrary constant of integration and allows for an infinite number of solutions that satisfy the differential equation.

## 4. Can the general solution be used to find a specific solution?

Yes, the general solution can be used to find a specific solution by substituting specific initial conditions into the equation. This will determine the value of the constant C and provide a unique solution.

## 5. Are there any other methods to find the general solution of a differential equation?

Yes, there are other methods such as the method of variation of parameters and the method of Laplace transforms. These methods can be used for more complex differential equations that cannot be solved using the method of undetermined coefficients.

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