Question

y^prime=((1)/(x)+x^x) y^x

Solution

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3 (286 Votos)

Silvana Elite · Tutor por 8 anos

Resposta

To solve the differential equation \( y' = \left(\frac{1}{x} + x^x\right) y^x \), we can use the method of separation of variables. Let's separate the variables

and

on different sides of the equation.First, rewrite the equation as:

Separate the variables by dividing both sides by

Now, integrate both sides:

The left side can be simplified using the power rule for integration:

For the right side, we need to integrate term by term:

The first term is straightforward:

The second term involves an exponential function with a variable base, which can be solved using the substitution method. Let

, then

, and \( du = (x \ln x + \ln x) \, dx \). However, this term is more complex and typically requires advanced techniques or numerical methods to solve.Combining the results, we get:

where

is the constant of integration.This is the general solution to the given differential equation.