# 2.1.2 Find the general solution of the given differential equation

• MHB
• karush
In summary, the general solution of the given differential equation is $\displaystyle y = -\frac{t^3}{3}+c_1e^{2t}$. To obtain $u(t)$, we use the integrating factor $\displaystyle u(t)=\exp\int -2 \, dx =e^{-2t}$ and multiply it through the equation. After simplifying and integrating, we get $\displaystyle e^{-2t}y=\int t^2\, dt=\frac{t^3}{3}+c_1$. Dividing by $e^{-2t}$ gives us the final solution of $\displaystyle y = -\frac{t^3}{3}+c karush Gold Member MHB Find the general solution of the given differential equation$\displaystyle y^\prime - 2y = t^2 e^{2t}$Obtain$u(t)\displaystyle u(t)=\exp\int -2 \, dx =e^{-2t}$Multiply thru with$e^{-2t}e^{-2t}y^\prime
+ 2e^{-2t}y
= t^2 $Simplify:$(e^{-2t}y)'= t^2$Integrate:$\displaystyle e^{-2t}y=\int t^2\, dt=-\frac{t^3}{3}+c_1$Divide thru by$e^{-2t}\displaystyle -\frac{t^3e^{2t}}{3}+c_1e^{2t}$ok took me 2 hours hope it ok any suggest? $$\tiny\textbf{Text: Elementary Differential Equations and Boundary Value Problems}$$ karush said: Find the general solution of the given differential equation$\displaystyle y^\prime - 2y = t^2 e^{2t}$Obtain$u(t)\displaystyle u(t)=\exp\int -2 \, dx =e^{-2t}$Multiply thru with$e^{-2t}e^{-2t}y^\prime
+ 2e^{-2t}y
= t^2 $Simplify:$(e^{-2t}y)'= t^2$Integrate:$\displaystyle e^{-2t}y=\int t^2\, dt=-\frac{t^3}{3}+c_1$Divide thru by$e^{-2t}\displaystyle -\frac{t^3e^{2t}}{3}+c_1e^{2t}\$

ok took me 2 hours hope it ok
any suggest?

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

Did you check your result? Seems as though the integral didn't quite happen correctly.

tkhunny said:
Did you check your result? Seems as though the integral didn't quite happen correctly.

$$\displaystyle e^{-2t}y=\int t^2\, dt=\frac{t^3}{3}+c_1$$

you must mean the negative sign?
which I took out

## 1. What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It describes how the rate of change of a variable is related to the value of the variable itself.

## 2. What is the general solution of a differential equation?

The general solution of a differential equation is the most general form of the equation that satisfies all possible solutions. It includes all possible solutions with arbitrary constants, so it is not a specific solution but a family of solutions.

## 3. How do you find the general solution of a differential equation?

To find the general solution of a differential equation, you need to solve the equation by integrating both sides and then adding an arbitrary constant to the solution. This constant will account for all possible solutions and will be determined by any initial conditions given in the problem.

## 4. Can a differential equation have multiple solutions?

Yes, a differential equation can have multiple solutions. The general solution of a differential equation includes all possible solutions, and the arbitrary constant in the solution can result in different specific solutions.

## 5. What is the importance of finding the general solution of a differential equation?

Finding the general solution of a differential equation allows us to understand the behavior and relationships between variables in a system. It also helps in solving real-world problems in various fields such as physics, engineering, and economics.

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