# Solve Differential Equation y"+3y'+2y=sin(e^x)

• chaose
In summary, the conversation is about finding the general solution to a differential equation using the variation of parameters method. The equations and steps involved in finding the particular solution are mentioned, including the use of a Wronskian and the integral of a function. The speaker also mentions a useful website for solving mathematical problems.
chaose

## Homework Statement

Integrate [e^(-x)*sin(e^x)]dx ?

This is part of the work. I was actually rying to find the general solution to the differential equation y"+3y'+2y=sin(e^x) using the variation of the parameter method.

## Homework Equations

Particular solution Y = u1(x)y1(x) + u2(x)y2(x), where u1 and u2 are functions of x, y1 and y2 were determined from the homogeneous version of the equation.

Wronskian = e^(-x)+2e^(-x) = 3e^(-x)

u1 = integral of -(e^x)sin(e^x)/(3e^-x)dx = -1/3 (sin(e^x)-e^xcos(e^x))

## The Attempt at a Solution

I don't know how to integrate u2. Is there a trick involved?

www.quickmath.com - powered by webMathematica: this site is very handy and will do algebra (factoring, simplify, partial fraction decomposition, expand), solve equations/inequalities (single or a system), plot equations/inequalities (single or a system), derivatives, definite or indefinite integrals, and do matrix arithmetic/inverses/determinants.

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It is used to model various physical phenomena in fields such as physics, engineering, and economics.

## 2. How do you solve a differential equation?

There are various methods to solve a differential equation depending on its type and order. Some common methods include separation of variables, substitution, and using integrating factors.

## 3. What is the order of this differential equation?

The order of a differential equation is determined by the highest derivative present in the equation. In this case, the order is 2 since the highest derivative is y".

## 4. How do you handle the non-homogeneous term in this equation?

In this equation, the non-homogeneous term is sin(e^x). To solve for the particular solution, we use the method of undetermined coefficients, where we assume a form for the solution and find the coefficients that satisfy the equation.

## 5. Can you explain the physical meaning of this differential equation?

This differential equation can represent a variety of physical phenomena, such as the motion of a damped harmonic oscillator or the decay of a radioactive substance. The term sin(e^x) can represent an external force or perturbation acting on the system.

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