An integrating factor of the D. E. : (1 + x^3)(dy / dx) + 3x^2y = x^2 is _____

Understand the Problem

The question is asking for the integrating factor of the given differential equation (D.E.). An integrating factor is a function that we multiply the entire differential equation by to make it easier to solve. We will identify the appropriate integrating factor from the options provided.

Answer

The integrating factor is $ \mu(x) = e^{\int P(x) \, dx} $.

Steps to Solve

Identify the form of the differential equation

First, we need to ensure that the differential equation is in the standard linear form, which is given by

$$ \frac{dy}{dx} + P(x)y = Q(x) $$

where ( P(x) ) and ( Q(x) ) are functions of ( x ).

Determine the integrating factor

The integrating factor ( \mu(x) ) is calculated using the formula:

$$ \mu(x) = e^{\int P(x) , dx} $$

So we need to integrate ( P(x) ).

Compute the integral

Perform the integration of ( P(x) ):

$$ \int P(x) , dx $$

This will give us the exponent we need in our integrating factor.

Formulate the integrating factor

Substitute the result from the integration into the formula for the integrating factor:

$$ \mu(x) = e^{(\text{result from integration})} $$

Now we have our integrating factor, which we can use for solving the differential equation.

The integrating factor is given by

$$ \mu(x) = e^{\int P(x) , dx} $$

after we compute the integral of ( P(x) ).

More Information

The integrating factor is crucial in transforming a non-exact differential equation into an exact one, making it solvable. The method is widely used in solving first-order linear ordinary differential equations.

Tips

Incorrectly identifying ( P(x) ): Ensure ( P(x) ) is extracted correctly from the standard form of the D.E.
Mistaking the integrating factor formula: Remember that ( \mu(x) ) is based on the exponential of the integral of ( P(x) ), not a direct multiplication or sum.

Thank you for voting!