Graphs with Gradient Proportional to One of the Coordinates – A Level Mathematics B (MEI) OCR Revision

Differential Equations Representing Graphs with Gradient Proportional to One of the Coordinates

Exponential graphs frequently appear in higher level mathematics, often in the context of differential equations where the rate of change is proportional to the current value - typically written as dy/dx = ky, where k is a constant.
In the differential equation dy/dx = ky, y denotes the dependent variable, x is the independent variable, and the constant k determines the graph’s shape. When k > 0, the exponential graph shows growth, while for k < 0, the graph shows decay.
One solution to the differential equation dy/dx = ky is y = Ce^kx, where C is the value of y when x = 0. This represents an exponential function where the gradient at any point is proportional to its y-coordinate.

For k > 0, the graph of y = Ce^kx rises upwards as x increases, asymptotic to the x-axis (y=0) as x -> -∞, and tends towards ∞ as x -> ∞. The gradient at any point on the graph is proportional to its y-coordinate and gets steeper as x increases.
For k < 0, the graph of y = Ce^kx falls towards the x-axis as x increases, starts at y = C when x = 0 and tends towards 0 as x -> ∞. The gradient at any point is proportional to the y-coordinate and gets less steep as x grows.

In solving problems, it’s crucial to identify the growth or decay factor in the situation at hand, and link this to the constant k.
Common examples might consider the population of bacteria growing exponentially, with k indicating the growth rate, or radioactive decay, where k is the rate of decay. In either case, identifying k is key.
To find the value of C, we often use an initial condition that provides the value of y at a specific x. From the general solution y = Ce^kx, we substitute the given values for x and y to find C.

When provided with an exponential equation and an additional condition, and tasked with determining the constants, it can be beneficial to use the logarithm properties to transform the exponential equation. This makes it simpler to solve for the desired constants.
For instance, to solve for k in the equation y = Ce^kx, after determining C using an initial condition, the equation could be rewritten in logarithmic form using the equation ln(y) = kx + ln(C). By substituting a second known x and y condition into this equation, k can be found.