Question

True or False? In Exercises \(89-92\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A slope field shows that the slope at the point \((1,1)\) is \(6 .\) This slope field represents the family of solutions for the differential equation \(y^{\prime}=4 x+2 y\)

Short answer

True

Step-by-step solution

Substitute the point into the differential equation

Substitute the coordinates of the given point \((1,1)\) into the equation \(y'=4x+2y\). This will give the value of \(y'\) (the slope of the tangent line to the solution curve at the given point), which should be equivalent to 6 according to the problem statement.

Evaluate and compare with the given slope

After substituting, the expression becomes \(y' = 4(1) + 2(1) = 6\). However, on evaluating this, it results in \(y'=6\), which indeed matches the given slope in the problem statement.

Decision about the truth of the statement

Since the slope at the point \((1,1)\) as evaluated from the given differential equation matches with that provided in the statement (both are 6), it can be concluded that the statement is true.

Key concepts

Slope Field

Slope fields, often referred to as direction fields, visually represent the slope of a differential equation at various points on a plane. They offer a way to predict the behavior of solutions without solving the differential equation directly. For an equation like \( y' = 4x + 2y \), a slope field shows tiny tangent lines at numerous points. Each small line's angle corresponds to the slope calculated by the differential equation.

• This graphical representation helps understand the general trend and direction of solutions.
• You plot lines based on the slopes obtained for given points.
• It provides insight into families of solutions without needing explicit solution forms.

They are particularly useful for visualizing differential equations with complex solutions or where analytical solutions are hard to obtain.

Solutions of Differential Equations

Differential equations define a relationship between a function and its derivatives. The "solution" to a differential equation is a function that satisfies the equation when substituted. Solving them involves finding functions \( y(x) \) that make the equation true.

• General solutions include arbitrary constants and represent a family of potential solutions.
• Particular solutions are obtained by fixing these constants, often using initial conditions.
• In some cases, solutions can be explicit, providing \( y \) as a direct function of \( x \).

This concept is critical because solutions show how dynamic systems evolve over time or space, based on their initial setup or boundary conditions.

Tangent Line

The tangent line at a point on a curve represents the curve's slope at that precise location, effectively approximating the function locally. For a differentiable function represented by a differential equation, this slope is given by the derivative.

• The equation for a tangent line at a point \((x_0, y_0)\) is \( y - y_0 = m(x - x_0) \), where \( m \) is the slope found from the derivative.
• In our example, substituting the point \((1,1)\) into \( y' = 4x + 2y \) gave a slope of 6, implying \( y - 1 = 6(x - 1) \).
• The tangent line offers linear approximation, useful for predicting near behavior.

Understanding and calculating tangent lines are crucial for analyzing local behaviors of functions represented by differential equations.

Substitution in Equations

Substitution is invaluable in solving differential equations because it simplifies evaluating and finding specific solutions by plugging in numbers or expressions. In our context, specifically substituting the point \((1,1)\) into the equation \( y' = 4x + 2y \) verifies the tangent slope of the solution curve at that point.

• First, identify variables and their values to substitute into the equation, like \( x = 1 \) and \( y = 1 \).
• Evaluate the equation to check consistency with provided conditions or results, as shown by determining \( y' = 6 \).
• This method confirms the accuracy of theoretical predictions, ensuring calculated solutions align with the points on the slope field.

By using substitution effectively, you can validate predictions and understand specific behaviors of differential equation solutions based on given conditions.