y’=14x
The equation y’ = 14x represents a first-order derivative
The equation y’ = 14x represents a first-order derivative. In mathematics, this notation represents the derivative of a function y with respect to x, or the rate of change of y with respect to x.
In this specific case, y’ = 14x means that the rate of change of y with respect to x is given by the expression 14x. This implies that as x changes, y changes at a rate that is proportional to x, with a constant of proportionality equal to 14.
To find the function y that satisfies this equation, we need to integrate both sides with respect to x. Integrating y’ = 14x gives:
∫y’ dx = ∫14x dx
y = 14∫x dx
Now, we can integrate the right hand side. The integral of x with respect to x is (x^2)/2. Therefore:
y = 14 * (x^2)/2 + C
where C represents a constant of integration. The constant C accounts for potential initial conditions or boundary conditions that are not given in the equation y’ = 14x.
In summary, the solution to the differential equation y’ = 14x is a family of functions given by y = (7/2) * x^2 + C, where C is the constant of integration.
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