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# Determine the general solution of the equation y′+y=e−x .

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## Answer to a math question Determine the general solution of the equation y′+y=e−x .

Hank
4.8
To find the general solution of the given differential equation, we can use the method of integrating factor.

Step 1: Write the equation in standard form.
The given equation is:
y' + y = e^$-x$

Step 2: Identify the coefficients of y' and y.
In this case, the coefficient of y' is 1 and the coefficient of y is 1.

Step 3: Determine the integrating factor.
The integrating factor $IF$, denoted by μ, is given by the formula:
μ = e^$∫p(x$dx)

where p$x$ is the coefficient of y in the standard form. In this case, p$x$ = 1.

∫p$x$dx = ∫1dx = x

Therefore, the integrating factor μ is:
μ = e^x

Step 4: Multiply the entire equation by the integrating factor.
By multiplying the equation y' + y = e^$-x$ by the integrating factor μ = e^x, we get:
e^x$y' + y$ = e^x$e^(-x$)

Simplifying the equation gives:
ye^x + e^xy' = 1

Step 5: Rewrite the left side of the equation.
Using the product rule of derivatives, we can rewrite the left side of the equation as:
$d/dx$$ye^x$ = 1

Step 6: Integrate both sides of the equation.
Integrating both sides of the equation gives:
∫$d/dx$$ye^x$ dx = ∫1 dx

Integrating the left side gives:
ye^x = x + C

where C is the constant of integration.

Step 7: Solve for y.
Dividing both sides of the equation by e^x gives:
y = $x + C$e^$-x$

Step 8: General solution.
The general solution of the differential equation is:
y = $x + C$e^$-x$

Answer: y = $x + C$e^$-x$

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