Question

How do you find and classify the critical points of this differential equation.
\[\dfrac{dx}{dt}=2x+4y+4\].

Answer 1

Hint: In this problem, we have to find the critical points of the differential equation. We can see that the given equation is already differentiated. We should know that the critical point occurs when the first derivative vanishes, i.e. \[\dfrac{dx}{dt}=0\]. We can substitute this in the given equation, to find the value of x, which is the critical point that fulfils the condition \[\dfrac{dx}{dt}=0\].

Complete step by step solution:
 We know that the given differential equation to which we have to find the critical point is
\[\dfrac{dx}{dt}=2x+4y+4\].
We should know that the critical point occurs when the first derivative vanishes, i.e. \[\dfrac{dx}{dt}=0\].
We can now substitute the above value in the equation, we get
\[\Rightarrow 2x+4y+4=0\]
We can now divide the above step by the number 2, we get
\[\Rightarrow \dfrac{2x}{2}+\dfrac{4y}{2}+\dfrac{4}{2}=0\]
We can now cancel the similar terms to get a simplified form, we get
\[\Rightarrow x+2y+2=0\]
We can now subtract 2y+2 on both the left-hand side and the right-hand side of the equation, we get
\[\Rightarrow x=-2y-2\]
Therefore, the critical point is x = -2y-2 for the given derivative \[\dfrac{dx}{dt}=2x+4y+4\].

Note:
Students should understand the concept of critical points, we should always remember that, a stationary point or a critical point in the domain of f such that \[f'\left( c \right)=0\] or \[f'\left( c \right)\] is undefined. So find \[f'\left( x \right)\] and look for the x values that make \[f'\] zero or undefined where f is still defined there.