Math Problem Statement

what is the integration facto of dy / dx + 1 / x y= x square

Solution

To solve the differential equation dydx+1xy=x2\frac{dy}{dx} + \frac{1}{x} y = x^2, we can identify that this is a first-order linear differential equation of the form:

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

Here:

  • P(x)=1xP(x) = \frac{1}{x}
  • Q(x)=x2Q(x) = x^2

Step 1: Finding the integrating factor

The integrating factor μ(x)\mu(x) is given by:

μ(x)=eP(x)dx=e1xdx\mu(x) = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx}

The integral of 1x\frac{1}{x} is lnx\ln |x|, so:

μ(x)=elnx=x\mu(x) = e^{\ln |x|} = |x|

Since xx is positive (due to the form of the problem), we can write:

μ(x)=x\mu(x) = x

Step 2: Multiply the entire equation by the integrating factor

Now, multiply the original differential equation by xx:

xdydx+xxy=xx2x \cdot \frac{dy}{dx} + \frac{x}{x} y = x \cdot x^2

Simplifying:

xdydx+y=x3x \frac{dy}{dx} + y = x^3

Step 3: Solve the equation

The left-hand side can now be written as a derivative:

ddx(xy)=x3\frac{d}{dx} (xy) = x^3

Now, integrate both sides with respect to xx:

xy=x3dxxy = \int x^3 dx

The integral of x3x^3 is:

xy=x44+Cxy = \frac{x^4}{4} + C

Step 4: Solve for yy

Now, divide by xx:

y=x34+Cxy = \frac{x^3}{4} + \frac{C}{x}

Thus, the solution to the differential equation is:

y=x34+Cxy = \frac{x^3}{4} + \frac{C}{x}

Would you like further clarification or more details on the process? Here are five related questions you might explore:

  1. How do we handle the integration factor for more complex functions?
  2. What happens if P(x)P(x) is a more complicated function, such as a trigonometric one?
  3. How does the integrating factor method compare to other methods for solving first-order linear differential equations?
  4. Can this method be extended to non-linear differential equations?
  5. How would the solution change if Q(x)Q(x) was exe^x instead of x2x^2?

Tip: Always verify the final solution by plugging it back into the original differential equation.

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Math Problem Analysis

Mathematical Concepts

First-Order Linear Differential Equations
Integrating Factors
Calculus

Formulas

First-order linear differential equation: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^∫P(x) dx

Theorems

Integrating Factor Method

Suitable Grade Level

Undergraduate Mathematics or Advanced High School Calculus