# Solving a First Order Differential Equation with Initial Conditions.

• uber_kim
In summary, the conversation discusses solving an initial value problem involving a linear equation and an integrating factor. The individual rearranges the equation and solves for the integrating factor before integrating both sides and finding the final solution. They also mention checking their answer with the book's answer and finding that it matches.
uber_kim

## Homework Statement

Solve the initial value problem:

t(dy/dt)+8y=t^3 where t>0 and y(1)=0

None?

## The Attempt at a Solution

It's a linear equation, so rearranged to dy/dt+8y/t=t^2.

Took the integrating factor e^(∫8/tdt)=t^8 and multiplied through.

(t^8)dy/dt+8t^7y=t^10.

∫(t^8y)'dt=∫t^10dt
t^8y=(t^11)/11 + C

Solve for C, I got -1/11.

Final solution is y=(t^3)/11 - 1/(11t^8)

This isn't right, though. Does anyone see where I made the mistake?

Thanks!

uber_kim said:

## Homework Statement

Solve the initial value problem:

t(dy/dt)+8y=t^3 where t>0 and y(1)=0

None?

## The Attempt at a Solution

It's a linear equation, so rearranged to dy/dt+8y/t=t^2.

Took the integrating factor e^(∫8/tdt)=t^8 and multiplied through.

(t^8)dy/dt+8t^7y=t^10.

∫(t^8y)'dt=∫t^10dt
t^8y=(t^11)/11 + C

Solve for C, I got -1/11.
Final solution is y=(t^3)/11 - 1/(11t^8)

This isn't right, though. Does anyone see where I made the mistake?

Hmm, strange. Maybe a typo. Thanks!

## 1. What is a first order differential equation?

A first order differential equation is an equation that involves a function and its first derivative. It is used to model various physical and mathematical phenomena in science and engineering.

## 2. What are initial conditions in a first order differential equation?

Initial conditions refer to the values of the function and its derivative at a specific point. These conditions are used to uniquely determine the solution to the differential equation.

## 3. How do you solve a first order differential equation with initial conditions?

To solve a first order differential equation with initial conditions, you can use various methods such as separation of variables, integrating factors, and substitution. The specific method used depends on the form of the equation.

## 4. What is the general solution to a first order differential equation with initial conditions?

The general solution to a first order differential equation with initial conditions is a family of solutions that satisfy the equation. It includes a constant of integration that is determined by the initial conditions.

## 5. Can you give an example of solving a first order differential equation with initial conditions?

Sure, let's consider the equation dy/dx = x + 2 with the initial condition y(0) = 1. Using separation of variables, we can rewrite the equation as dy = (x + 2)dx. Integrating both sides gives us y = (x^2/2) + 2x + C. Plugging in the initial condition, we get C = -1. Therefore, the solution to the differential equation with the given initial condition is y = (x^2/2) + 2x - 1.

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