# How can i solve this differencial equation? : #y'+x^2 y=x^2#

# y = 1 + Ce^(-1/3x^3) #

When we have a First Order Linear non-homogeneous Ordinary Differential Equation of the following form, we can use an integrating factor;

We have:

Our original ODE has now become a Separable ODE as a result of which we can "separate the variables" to obtain::

It is easy to integrate this, and we obtain:

Finally, arriving at the clear General Solution:

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A simpler method to the one given in the other answer

This ODE is separable.

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To solve the differential equation y' + x^2y = x^2, you can use an integrating factor approach. First, rearrange the equation into the standard form for linear first-order differential equations, which is y' + P(x)y = Q(x), where P(x) is the coefficient of y and Q(x) is the function on the right side of the equation. Then, determine the integrating factor, which is e^(∫P(x)dx). In this case, P(x) = x^2, so the integrating factor is e^(∫x^2 dx). Next, multiply both sides of the equation by the integrating factor. This gives you the left side as (e^(∫x^2 dx)) * y' + (e^(∫x^2 dx)) * x^2 * y. Now, recognize that the left side is the derivative of (e^(∫x^2 dx)) * y with respect to x. Integrate both sides of the equation with respect to x to find the general solution. Finally, you may need to apply initial conditions if they are provided to find the particular solution.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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