Math  /  Calculus

QuestionTime left 1:25:34
Determine the integrating factor then solve the differential equation.
Question 1 Not yet answered Points out of 10.00 Flagquestion dydx2yx=3x3\frac{d y}{d x}-\frac{2 y}{x}=3 x^{3}

Studdy Solution

STEP 1

What is this asking? We're given a differential equation and need to find a magical *integrating factor* to make it solvable, and then actually solve it for yy in terms of xx. Watch out! Don't mix up the steps for finding the integrating factor and solving the equation.
Also, be careful with those exponents when integrating!

STEP 2

1. Find the Integrating Factor
2. Multiply and Rewrite
3. Integrate Both Sides
4. Solve for yy

STEP 3

Our differential equation is in the form dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x), where P(x)=2xP(x) = -\frac{2}{x} and Q(x)=3x3Q(x) = 3x^3.
The integrating factor, let's call it μ(x)\mu(x), is like a key to unlock this equation!

STEP 4

To find this magical key, we use the formula μ(x)=eP(x)dx\mu(x) = e^{\int P(x) dx}.
So, let's **calculate** that integral: P(x)dx=2xdx=21xdx=2lnx\int P(x) dx = \int -\frac{2}{x} dx = -2 \int \frac{1}{x} dx = -2 \ln|x| Remember, the integral of 1x\frac{1}{x} is lnx\ln|x|!

STEP 5

Now, plug this back into our **integrating factor formula**: μ(x)=e2lnx=elnx2=x2=1x2\mu(x) = e^{-2 \ln|x|} = e^{\ln|x|^{-2}} = |x|^{-2} = \frac{1}{x^2} We can drop the absolute value because we'll be multiplying both sides of the equation by the integrating factor, so any negative signs will cancel out.

STEP 6

Now, let's **multiply** both sides of our original differential equation by our shiny new integrating factor, 1x2\frac{1}{x^2}: 1x2dydx1x22yx=1x23x3\frac{1}{x^2} \cdot \frac{dy}{dx} - \frac{1}{x^2} \cdot \frac{2y}{x} = \frac{1}{x^2} \cdot 3x^3 1x2dydx2yx3=3x\frac{1}{x^2} \frac{dy}{dx} - \frac{2y}{x^3} = 3x

STEP 7

Notice that the left side is now the **derivative** of a product!
Specifically, it's the derivative of yx2\frac{y}{x^2} with respect to xx.
We can rewrite the equation as: ddx(yx2)=3x\frac{d}{dx}\left(\frac{y}{x^2}\right) = 3x This is the magic of the integrating factor!

STEP 8

Let's **integrate** both sides with respect to xx: ddx(yx2)dx=3xdx\int \frac{d}{dx}\left(\frac{y}{x^2}\right) dx = \int 3x dx yx2=3x22+C\frac{y}{x^2} = \frac{3x^2}{2} + CDon't forget the constant of integration, CC!

STEP 9

Finally, let's **isolate** yy: y=x2(3x22+C)=3x42+Cx2y = x^2 \left(\frac{3x^2}{2} + C\right) = \frac{3x^4}{2} + Cx^2 And there we have it!

STEP 10

The solution to the differential equation is y=3x42+Cx2y = \frac{3x^4}{2} + Cx^2.

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