# How Do You Solve the Initial Value Problem $y'+5y=0$ with $y(0)=2$?

• MHB
• karush
In summary, we are given a first-order linear differential equation and an initial value to solve for. After applying the integrating factor and solving for $y$, we arrive at the solution $y=2e^{-5t}$.
karush
Gold Member
MHB
$\tiny{1.5.7.19}$
\nmh{157}
Solve the initial value problem
$y'+5y=0\quad y(0)=2$
$u(x)=exp(5)=e^{5t+c_1}$?

so tried
$\dfrac{1}{y}y'=-5$
$ln(y)=-5t+c_1$

apply initial values
$ln(y)=-5t+ln(2)\implies ln\dfrac{y}{2}=-5t \implies \dfrac{y}{2}=e^{-5t} \implies y=2e^{-5t}$

Last edited:
We will return to your problem after a short review of what input of a function means. Let's suppose that $f(x)=x+3$. What is $f(3)$?

6

here $u(t)=exp(t)= e^{\int t \ dt}$

Surely you mean $u(t) = \mathrm{e}^t$, not $\mathrm{e}^{\int{t}\,dt}$...

In actuality, the integrating factor is $u(t) = \mathrm{e}^{5\,t}$...

## What is a differential equation?

A differential equation is an equation that relates a function with its derivatives. It is used to describe various physical phenomena and is an essential tool in mathematics and science.

## Why is solving a differential equation important?

Solving a differential equation allows us to understand the behavior of a system over time. It helps us make predictions and model real-world problems in fields such as physics, engineering, and economics.

## What are the different methods for solving a differential equation?

There are several methods for solving a differential equation, including separation of variables, substitution, and using integrating factors. The choice of method depends on the type and complexity of the equation.

## Can all differential equations be solved analytically?

No, not all differential equations can be solved analytically. Some equations are too complex to have a closed-form solution, and numerical methods must be used to approximate the solution.

## How can differential equations be applied in real life?

Differential equations have many practical applications, such as predicting the growth of populations, modeling the spread of diseases, and analyzing the behavior of electrical circuits. They are also used in fields like finance, biology, and chemistry.

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