39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form
a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t). 
Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems. 


t³y′(t) + 3t²y = (1 + t)/t, y(1) = 6

Verified step by step guidance

1. Recognize that the given differential equation is of the form \(a(t)y'(t) + a'(t)y(t) = f(t)\), where \(a(t) = t^3\). This means the left side can be expressed as the derivative of the product \(a(t)y(t)\), i.e., \(\frac{d}{dt}(t^3 y(t))\).
2. Rewrite the equation using this product rule form: \(\frac{d}{dt}(t^3 y(t)) = \frac{1 + t}{t}\).
3. Integrate both sides with respect to \(t\) to find \(t^3 y(t)\): \(\int \frac{d}{dt}(t^3 y(t)) \, dt = \int \frac{1 + t}{t} \, dt\).
4. Simplify the integral on the right side by splitting the fraction: \(\int \left( \frac{1}{t} + 1 \right) dt = \int \frac{1}{t} dt + \int 1 dt\).
5. After integrating, solve for \(y(t)\) by dividing both sides by \(t^3\), then use the initial condition \(y(1) = 6\) to find the constant of integration.

Key Concepts

First-Order Linear Differential Equations

These are differential equations of the form y' + p(t)y = q(t), where p and q are functions of t. They can often be solved using integrating factors or by recognizing patterns that simplify the equation. Understanding their structure is essential for applying appropriate solution methods.

Product Rule and Recognizing Derivatives of Products

The product rule states that d/dt [a(t)y(t)] = a(t)y'(t) + a'(t)y(t). Recognizing when the left side of a differential equation matches this derivative allows rewriting the equation in a simpler form, facilitating direct integration to find solutions.

Initial Value Problems and Integration

An initial value problem specifies the value of the solution at a particular point, enabling determination of the integration constant after solving the differential equation. Integrating both sides with respect to t and applying the initial condition yields the unique solution.