Consider the initial value problem: x=dx/dt=0.5(x^2-2x), x(0)=c. Find the solution, domain and limits of the following scenarios: c=-1, c=1, c=3.
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First-order autonomous differential equations
The first-order autonomous differential equation dxdt=f(x)dxdt=f(x) is potentially the easiest of all differential equations. Notice that the independent variable tt does not explicitly appear in the equation ('autonomous') and we only have a first derivative and no higher derivatives ('first-order').
This equation is separable so we group variables and integrate:
dxdt=f(x)⟹1f(x)dx=dt⟹∫1f(x)dx=∫dt⟹G(x)=t+C⟹x=G−1(t+C)dxdt=f(x)⟹1f(x)dx=dt⟹∫1f(x)dx=∫dt⟹G(x)=t+C⟹x=G−1(t+C)
where CC is an arbitrary constant and G(x)G(x)is an antiderivative of 1f(x)1f(x).
In its most proper form we should write the solution explicitly as a function of tt: x(t)=G−1(t+C)x(t)=G−1(t+C).
Of course, the value of CC can be determined only when an initial value of x(t)x(t) is given.
Restrictions on domain of a solution arise when the original differential equation or the solution is not defined. We choose the largest possible interval containing our original value to be the domain.
The first-order autonomous differential equation dxdt=f(x)dxdt=f(x) is potentially the easiest of all differential equations. Notice that the independent variable tt does not explicitly appear in the equation ('autonomous') and we only have a first derivative and no higher derivatives ('first-order').
This equation is separable so we group variables and integrate:
dxdt=f(x)⟹1f(x)dx=dt⟹∫1f(x)dx=∫dt⟹G(x)=t+C⟹x=G−1(t+C)dxdt=f(x)⟹1f(x)dx=dt⟹∫1f(x)dx=∫dt⟹G(x)=t+C⟹x=G−1(t+C)
where CC is an arbitrary constant and G(x)G(x)is an antiderivative of 1f(x)1f(x).
In its most proper form we should write the solution explicitly as a function of tt: x(t)=G−1(t+C)x(t)=G−1(t+C).
Of course, the value of CC can be determined only when an initial value of x(t)x(t) is given.
Restrictions on domain of a solution arise when the original differential equation or the solution is not defined. We choose the largest possible interval containing our original value to be the domain.
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