# Basic Diff Eq Questions

• dillonmhudson
In summary, the conversation is about solving initial value problems and plotting their solutions for several values of y_0. The equations provided are dy/dt=-y+5, dy/dt=-2y+5, and dy/dt=-2y+10 with initial conditions y(0)=y_0. The attempt at a solution involves finding the general solutions for each equation and using the method of undetermined coefficients to find the solution for the equation y'+2y=2t+3sin(t). The solutions manual provides Y(t)=At+B+Ccost(t)-Dsin(t) as the solution, but the person is confused about why it is Ccos(t)-Dsin(t) instead of Ccos(t)+Dsin
dillonmhudson

## Homework Statement

Solve each of the following initial value problems and plot solutions for several values of y_0.
a) dy/dt=-y+5, y(0)=y_0
b) dy/dt=-2y+5, y(0)=y_0
c) dy/dt=-2y+10, y(0)=y_0

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## The Attempt at a Solution

I have solved all of the equations for general solutions but am having trouble plotting them (these are correct in the back of the book)
a) y=5+(y_0-5)*e^t
b) y=5/2+(y_0-5/2)*e^2t
c) y=5+(y_0-5)*e^2t

TIA

1. Homework Statement
Use method of undetermined coefficients to find the general solution of the given equation:
y'+2y=2t+3sin(t)
y=ce^-2t+Y(t) (where Y(t) is picked)
Solutions manual says: Y(t)=At+B+Ccost(t)-Dsin(t)

so that,
y=ce^-2t+At+B+Ccos(t)-Dsin(t)
I understand the "At+B" but why is it "Ccos(t)-Dsin(t)" and why is there a minus?

Thanks again

## 1. What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It is used to describe the relationship between a quantity and its rate of change.

## 2. What is the difference between an ordinary and partial differential equation?

An ordinary differential equation involves a single independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are used to describe phenomena in one dimension, while partial differential equations are used for phenomena in multiple dimensions.

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

The most common methods for solving differential equations include separation of variables, using an integrating factor, and using power series. Other methods include Laplace transforms, numerical methods, and similarity solutions.

## 4. What are initial and boundary conditions in a differential equation?

Initial conditions specify the values of the unknown function and its derivatives at a specific point in the domain. Boundary conditions specify the behavior of the function at the boundaries of the domain. Both initial and boundary conditions are necessary for finding a unique solution to a differential equation.

## 5. What are some real-life applications of differential equations?

Differential equations are used in many fields, such as physics, engineering, economics, and biology. They can be used to model and predict the behavior of systems that involve rates of change, such as population growth, heat transfer, and electrical circuits.

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