Question

Draw the direction field for the following differential equations, then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field? $$ y^{\prime}=t^{3} $$

Short answer

Yes, the solution curve \( y = \frac{t^4}{4} + C \) aligns with the direction field arrows.

Step-by-step solution

Understand the Direction Field

A direction field, or slope field, is a visual representation of a first-order differential equation. Each line segment in the field has a slope corresponding to the derivative at that point, given by the differential equation. For the equation \( y' = t^3 \), the slope of the tangent lines at any point (t, y) is \( t^3 \).

Create the Direction Field

To draw the direction field, consider various values of \( t \). Since the derivative \( y' = t^3 \) depends only on \( t \) and not on \( y \), for any fixed \( t \), the slope is constant regardless of \( y \). Calculate the slope at different \( t \) values, such as \( t = -2, -1, 0, 1, 2 \). Plot these slopes on a \( t-y \) plane by drawing short line segments at these points.

Identify the Pattern

Evaluate \( t^3 \) at different \( t \) values: - \( t = -2 \), slope is \(-8\)- \( t = -1 \), slope is \(-1\)- \( t = 0 \), slope is \(0\)- \( t = 1 \), slope is \(1\)- \( t = 2 \), slope is \(8\).The direction field will show horizontal line segments at \( t = 0 \) and increasing steepness as \( t \) moves away from zero.

Solve the Differential Equation

To solve the equation \( y' = t^3 \), integrate with respect to \( t \). \[ \int \, dy = \int t^3 \, dt \]\[ y = \frac{t^4}{4} + C \]where \( C \) is the constant of integration.

Sketch the Solution on the Direction Field

Draw the solution curve \( y = \frac{t^4}{4} + C \) over the direction field. This curve will follow the general direction of the arrows in the field. It should go through points that match the calculated slopes and follow the increase or decrease in steepness as indicated by the field.

Verify the Solution Matches the Direction Field

Examine whether the solution curve aligns with the arrows in the direction field. Each point on the curve should tangentially match the slope at that point as determined by \( t^3 \). This confirms that the solution follows the direction of the arrows.

Key concepts

Understanding the Direction Field

A direction field, often called a slope field, is a powerful tool that helps us visualize first-order differential equations. It consists of numerous small line segments, each representing the slope given by the differential equation at a specific point. In simpler terms, each segment is like a tiny arrow pointing in the direction a solution curve would go through that point.
Let's consider the equation \( y' = t^3 \). Here, the slope at any point \((t, y)\) is determined solely by the value of \(t\). Hence, this field's lines are independent of the \(y\)-value, and for any fixed \(t\), the slope remains the same. To visualize this, calculate the slope for several \(t\) points and represent them on a graph with short line segments. This creates a field where horizontal segments are shown at \(t = 0\) and the line steepness increases as \(t\) moves towards positive or negative values.
Direction fields help us anticipate the behavior of solutions without solving the equation analytically.

Creating a Slope Field

To create a slope field for the differential equation \( y' = t^3 \), we first need to identify the slopes at several points. Since \( y' \) is dependent only on \( t \), we calculate \( t^3 \) at different values of \( t \). Let's explore a few:

• For \( t = -2 \), the slope is \(-8\).
• For \( t = -1 \), the slope is \(-1\).
• For \( t = 0 \), the slope is \(0\).
• For \( t = 1 \), the slope is \(1\).
• For \( t = 2 \), the slope is \(8\).

With these calculations, we draw short line segments on the \(t-y\) coordinate plane. Each segment's inclination matches the calculated slope for its respective \( t \) value. This pattern of lines visually depicts the family of solutions to the differential equation. The slope field intuitively guides us in sketching potential solution curves, revealing how they should behave based on the calculated slopes.

Integration for Solving Differential Equations

Integration is a crucial step in solving differential equations. It transforms an expression containing a derivative into an expression that doesn't, revealing the function we are seeking. For our differential equation \( y' = t^3 \), we aim to determine \( y \) by integrating both sides with respect to \( t \).
Perform the integration to obtain: \[\int \, dy = \int t^3 \, dt\]which results in: \[y = \frac{t^4}{4} + C\]where \( C \) represents the constant of integration, accounting for the unknown starting condition of the integral.
Once calculated, this solution can be graphed over the direction field. The curve \( y = \frac{t^4}{4} + C \) should naturally align with the arrows in the direction field, providing a visual confirmation that the mathematical solution matches the graphical behavior.