91Ó°ÊÓ

When solving first-order linear differential equations, the integrating factor method is very useful. This method helps to simplify the equation so we can solve it more easily. The integrating factor, typically denoted as IF or \(\text{IF}\), converts the differential equation into a form that allows straightforward integration.
To find the integrating factor, we often use the formula: \[\text{IF} = e^{\bigg(\text{integral of}\frac{P(t)}{\text{where P(t) is the coefficient of y in the standard form}}\bigg)}\]
In our example, the coefficient of y in the standard form is \(\frac{1}{t}\). Thus,
\(\text{IF} = e^{\bigg(\text{ln}|t|\bigg)} = t\)
Multiplying the whole differential equation by this integrating factor transforms the equation into a much simpler form, enabling us to solve it by integrating both sides.

To solve differential equations using the integrating factor method, it’s important to first convert the given equation to its standard form. The standard form for a first-order linear differential equation is:
\(\frac{d}{dt} y + P(t)y = Q(t)\)
Here, \(\frac{d}{dt} y\) represents the derivative of y with respect to t, P(t) is a function of t multiplying y, and Q(t) is another function of t.
In our exercise, we start with: \(\text\big(t y' + y = t^2\big)\).
We rewrite it as:
\(\bigg( y' + \frac{1}{t} y = t \bigg)\)
Now our equation fits the standard form with \(\frac{1}{t}\) as P(t) and \(t\) as Q(t).

After finding our integrating factor and converting our differential equation into the standard form, we move on to solving it.

• First, we multiply through by the integrating factor. In our example, multiplying by t gives us:
• \(\frac{d}{dt} (t y) = t^3 \).

The left-hand side now looks like a product rule for differentiation:
\(\frac{d}{dt} \big( t y \big)\). Recognizing this, we can now integrate both sides with respect to t:
\(\bigg(t y = \frac{t^4}{4} + C\bigg)\)
Finally, we solve for y to find:
\(\bigg(y = \frac{t^3}{4} + \frac{C}{t}\bigg)\)
This solution \(\bigg(y\big(t\big)\) is our general solution for the given differential equation. Integrating properly and carefully helps attain the most accurate function for y.
Remember, always double-check your work. Understanding each step and how they interconnect makes solving differential equations more manageable.