Math / Calculus

Question5. Find the solution to differential equations $\frac{d y}{d x}+\frac{2 y}{x}=\frac{\sin (x)}{x^{2}}$

Studdy Solution

STEP 1

What is this asking? We're trying to find a function $y$ of $x$ that satisfies the given differential equation.
It's like a puzzle where we have a relationship between the function and its derivative, and we need to figure out what the function is! Watch out! Don't forget to consider the integrating factor, it's the key to unlocking this type of problem!
Also, be careful with your integration and algebra.

STEP 2

1. Identify the Form
2. Find the Integrating Factor
3. Multiply and Integrate
4. Solve for $y$

STEP 3

Alright, let's recognize that this differential equation is in the form $\frac{dy}{dx} + P(x)y = Q(x)$ , which is a first-order linear differential equation.
Here, $P(x) = \frac{2}{x}$ and $Q(x) = \frac{\sin(x)}{x^2}$ .
Knowing this form is super important because it tells us exactly how to approach the problem!

STEP 4

The integrating factor is like a magic key!
It's given by $e^{\int P(x) dx}$ .
So, let's calculate it: $e^{\int \frac{2}{x} dx} = e^{2 \ln|x|} = e^{\ln(x^2)} = x^2$ Remember, we're using the property that $a \ln(b) = \ln(b^a)$ and $e^{\ln(c)} = c$ .
We can drop the absolute value since we'll be multiplying both sides of the equation by the integrating factor.

STEP 5

Now, we multiply both sides of the original differential equation by our integrating factor, $x^2$ : $x^2 \frac{dy}{dx} + x^2 \cdot \frac{2y}{x} = x^2 \cdot \frac{\sin(x)}{x^2}$ This simplifies to: $x^2 \frac{dy}{dx} + 2xy = \sin(x)$

STEP 6

Notice that the left side is the derivative of $x^2y$ with respect to $x$ .
That's the whole point of the integrating factor!
So, we can rewrite the equation as: $\frac{d}{dx}(x^2y) = \sin(x)$

STEP 7

Now, we integrate both sides with respect to $x$ : $\int \frac{d}{dx}(x^2y) dx = \int \sin(x) dx$ This gives us: $x^2y = -\cos(x) + C$ where $C$ is the constant of integration.
Don't forget that constant!

STEP 8

Finally, we solve for $y$ by dividing both sides by $x^2$ : $y = \frac{-\cos(x) + C}{x^2}$ And there we have it!

STEP 9

The solution to the differential equation is $y = \frac{-\cos(x) + C}{x^2}$ .

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