Question

True or False Any two solutions to the differential equation \(d v / d x=5\) are parallel lines. Justify your answer.

Short answer

True, any two solutions to the differential equation \(d v / d x=5\) are parallel lines.

Step-by-step solution

Understanding the Differential Equation

Given the differential equation \(d v / d x=5\). This is a first order and a linear differential equation. The given differential equation states how the derivative of v with respect to x is constant and equals to 5. The function v(x) can be understood as a line in the x-y plane, and dv/dx represents the slope of that line. Since dv/dx equals the constant 5, we know that the line must slope upwards at an angle whose tangent is 5.

Finding the General Solution

Generally, the solution to a differential equation of the form \( dv / dx = a \), where a is a constant, is \( v = ax + b \). Here, a is 5. Therefore, the general solution is \( v = 5x + b \). Here, b is an arbitrary constant and represent the y-intercept of v = 5x + b.

Analyzing the Solutions

In the general solution \( v = 5x + b \), it is clear that 5 is the slope of the line. So any function of the type \( v = 5x + b \) is a straight line with a slope of 5. As the slope is the same for any line of this type, these lines are parallel to each other. As such, it can be stated that any two solutions to the given differential equation are parallel lines.