# 2.1.7 DE y'e -2ty =2te^(-t^2)

• MHB
• karush
Multiplication is commutative. Division is not. But very close to commutative. In summary, the general solution of the given differential equation is $y=c_1 e^{t^2}-\frac{e^{-t^2}}{2}$, obtained by multiplying through with $e^{-t^2}$, simplifying and integrating, then dividing by $e^{-t^2}$.
karush
Gold Member
MHB
Find the general solution of the given differential equation
$\displaystyle y^\prime -2ty =2te^{-t^2}\\$
Obtain $u(t)$
$\displaystyle u(t)=\exp\int -2t \, dt =e^{-t^2}$$%e^(-t^2)\\ Multiply thru with e^{-t^2} (e^{-t^2})y^\prime -(e^{-t^2})2ty =(e^{-t^2})2te^{-t^2}\\ Simplify: ((e^{-t^2})y)'= 2te^{-2t^2}\\ Integrate: \displaystyle e^{-t^2}y=\int 2te^{-2t^2} dt =-\frac{ e^{-2t^2}}{2}+c_1\\ Divide by e^{-t^2} \displaystyle y=-\frac{e^{-t^2}}{2}+c_1 e^{t^2}\\ Answer from \textbf{W|A} \displaystyle y=\color{red}{c_1 e^{t^2}-\frac{e^{-t^2}}{2}} ok got to be some typos in this otherwise suggestions$$\tiny\textsf{Text Book: Elementary Differential Equations and Boundary Value Problems}$$Last edited: karush said: Find the general solution of the given differential equation \displaystyle y^\prime -2ty =2te^{-t^2}\\ Obtain u(t) \displaystyle u(t)=\exp\int -2t \, dt =e^{-t^2}$$%e^(-t^2)\\$
Multiply thru with $e^{-t^2}$
$(e^{-t^2})y^\prime -(e^{-t^2})2ty =(e^{-t^2})2te^{-t^2}\\$
Simplify:
$((e^{-t^2})y)'= 2te^{-2t^2}\\$
Integrate:
$\displaystyle e^{-t^2}y=\int 2te^{-2t^2} dt =-\frac{ e^{-2t^2}}{2}+c_1\\$
Divide by $e^{-t^2}$
$\displaystyle y=-\frac{e^{-t^2}}{2}+c_1 e^{t^2}\\$
Answer from $\textbf{W|A}$
$\displaystyle y=\color{red}{c_1 e^{t^2}-\frac{e^{-t^2}}{2}}$

ok got to be some typos in this
otherwise suggestions$$\tiny\textsf{Text Book: Elementary Differential Equations and Boundary Value Problems}$$

Where do you believe those two results are different?

order of terms

karush said:
order of terms

Which is nothing. Addition is commutative.

## 1. What is the significance of the numbers in "2.1.7 DE y'e -2ty =2te^(-t^2)"?

The numbers in this equation represent the coefficients and variables used in a differential equation. The specific values may vary depending on the problem being solved.

## 2. How is this differential equation solved?

This equation can be solved using various techniques such as separation of variables, integrating factors, or using a power series method. The specific method used depends on the type and complexity of the equation.

## 3. What does the term "DE" stand for in "2.1.7 DE y'e -2ty =2te^(-t^2)"?

The term "DE" stands for "differential equation." A differential equation is an equation that contains derivatives of one or more dependent variables with respect to one or more independent variables.

## 4. How is this differential equation used in scientific research?

Differential equations are used in various fields of science to model and analyze systems that involve change over time. They can be used to study physical phenomena, biological processes, and many other systems.

## 5. Can this differential equation be applied to real-world problems?

Yes, this particular differential equation can be used to model and solve various real-world problems such as population growth, chemical reactions, and circuit analysis. However, the specific values and variables may vary depending on the problem being solved.

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