Linear Differential Equation - Initial Value Problem

In summary, the initial value problem asks to solve an equation with an integration factor of μ(x) = ex-x2. After multiplying both sides of the equation by the integration factor, the resulting equation is different from the one in the solution. However, upon further examination, it is realized that the left side of the equation is equal to the derivative of ex-x2⋅y, simplifying the problem.
  • #1
thaalescosta
11
0
Hello, I'm struggling with a simple problem here.

It asks me to solve the following initial value problem:
fSfRWNn.png


So far I've calculated the integration factor μ(x) = ex-x2 and I multiplied both sides of the equation by it and got this:
http://www4c.wolframalpha.com/Calculate/MSP/MSP36171f59839bdf0g6cbd000044defig6e40h0ei0?MSPStoreType=image/gif&s=64&w=256.&h=47.

The problem I'm having is that the equation I got is different than the equation in the first line of the solution:
2zu70dx.png


I don't understand how the ex-x2⋅(1 - 2x)⋅y disappearedEDIT: Actually, nevermind. My brain was too lazy to realize that the left part of the equation is equal to the derivative of ex-x2⋅y so I could just simplify it
​
 
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  • #2
Well done.
 

Related to Linear Differential Equation - Initial Value Problem

1. What is a linear differential equation?

A linear differential equation is a mathematical equation that involves the derivatives of a dependent variable with respect to an independent variable. The variables in a linear differential equation are related by linear functions, making the equation linear.

2. What is an initial value problem?

An initial value problem is a type of differential equation that involves finding the solution for a dependent variable at a specific initial value of the independent variable. In other words, it is a differential equation that includes an initial condition.

3. How do you solve a linear differential equation?

To solve a linear differential equation, you can use methods such as separation of variables, integrating factors, or variation of parameters. The specific method used will depend on the type of equation and the given initial conditions.

4. What is the importance of solving an initial value problem?

Solving an initial value problem allows us to find the specific solution for a differential equation at a specific point, rather than a general solution. This is important in many real-world applications where we need to know the exact value of a dependent variable at a given time or condition.

5. Can a linear differential equation have multiple solutions?

Yes, a linear differential equation can have multiple solutions. This is because the general solution of a linear differential equation includes an arbitrary constant, which can take on different values depending on the initial conditions. However, the number of solutions will always be equal to the order of the differential equation.

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