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

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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.
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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?
 
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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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