Can Someone Tell The Approach.
Introduction
In the realm of calculus, we often encounter functions that are intertwined in complex ways. The given problem presents two differentiable functions, f(x) and g(x), with specific conditions that need to be satisfied. Our goal is to find the functions f(x) and g(x) that meet these conditions. In this article, we will delve into the world of integration and exponential functions to uncover the solution.
Understanding the Conditions
The problem provides three conditions that f(x) and g(x) must satisfy:
(i) Initial Values: f(0) = 2 and g(0) = 1. These conditions give us a starting point to work with.
(ii) Relationship between f'(x) and g(x): f'(x) = g(x). This condition establishes a direct relationship between the derivatives of f(x) and g(x).
(iii) Relationship between g'(x) and f(x): g'(x) = f(x). This condition further connects the derivatives of f(x) and g(x).
Breaking Down the Conditions
Let's analyze each condition separately to gain a deeper understanding of the problem.
Condition (i): Initial Values
The initial values of f(x) and g(x) are given as f(0) = 2 and g(0) = 1. These values provide a starting point for our solution.
Condition (ii): Relationship between f'(x) and g(x)
The condition f'(x) = g(x) implies that the derivative of f(x) is equal to g(x). This means that the rate of change of f(x) is directly related to g(x).
Condition (iii): Relationship between g'(x) and f(x)
The condition g'(x) = f(x) implies that the derivative of g(x) is equal to f(x). This means that the rate of change of g(x) is directly related to f(x).
Finding the Functions
To find the functions f(x) and g(x), we need to use the given conditions and the properties of derivatives.
Using the Chain Rule
We can use the chain rule to differentiate the functions f(x) and g(x). The chain rule states that if we have a composite function of the form h(x) = f(g(x)), then the derivative of h(x) is given by h'(x) = f'(g(x)) * g'(x).
Applying the Chain Rule to f(x) and g(x)
Let's apply the chain rule to f(x) and g(x). We have:
f'(x) = g(x) g'(x) = f(x)
Using the chain rule, we can write:
f''(x) = g'(x) * g'(x) = f(x) * f(x) = (f(x))^2 g''(x) = f'(x) * f'(x) = g(x) * g(x) = (g(x))^2
Solving the Differential Equations
We now have two differential equations:
f''(x) = (f(x))^2 g''(x) = (g(x))^2
These differential equations can be solved using standard techniques.
Solving the Differential Equation for f(x)
To solve the differential equation f''(x) = (f(x))^2, we can use the following substitution:
u = f(x)
Then, we have:
u' = f'(x) u'' = f''(x)
Substituting u'' = (u)^2, we get:
u'' = (u)^2
This is a separable differential equation, which can be solved as follows:
∫(1/(u^2)) du = ∫dx -1/u = x + C
where C is a constant.
Solving the Differential Equation for g(x)
To solve the differential equation g''(x) = (g(x))^2, we can use the following substitution:
v = g(x)
Then, we have:
v' = g'(x) v'' = g''(x)
Substituting v'' = (v)^2, we get:
v'' = (v)^2
This is a separable differential equation, which can be solved as follows:
∫(1/(v^2)) dv = ∫dx -1/v = x + D
where D is a constant.
Finding the Functions f(x) and g(x)
Now that we have solved the differential equations, we can find the functions f(x) and g(x).
From the solution to the differential equation for f(x), we have:
-1/u = x + C u = -1/(x + C)
Substituting u = f(x), we get:
f(x) = -1/(x + C)
Similarly, from the solution to the differential equation for g(x), we have:
-1/v = x + D v = -1/(x + D)
Substituting v = g(x), we get:
g(x) = -1/(x + D)
Using the Initial Conditions
We can now use the initial conditions to find the values of C and D.
From the initial condition f(0) = 2, we have:
f(0) = -1/(0 + C) = 2 C = -1/2
Similarly, from the initial condition g(0) = 1, we have:
g(0) = -1/(0 + D) = 1 D = -1
Finding the Final Answer
Now that we have found the values of C and D, we can find the final answer.
f(x) = -1/(x - 1/2) g(x) = -1/(x + 1)
Therefore, the functions f(x) and g(x) that satisfy the given conditions are:
f(x) = -1/(x - 1/2) g(x) = -1/(x + 1)
Conclusion
In this article, we have solved the problem of finding two interconnected functions f(x) and g(x) that satisfy specific conditions. We have used the properties of derivatives, the chain rule, and standard techniques to solve the differential equations. The final answer is:
f(x) = -1/(x - 1/2) g(x) = -1/(x + 1)
Introduction
In our previous article, we solved the problem of finding two interconnected functions f(x) and g(x) that satisfy specific conditions. We used the properties of derivatives, the chain rule, and standard techniques to solve the differential equations. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solution.
Q: What are the conditions that f(x) and g(x) must satisfy?
A: The conditions that f(x) and g(x) must satisfy are:
(i) f(0) = 2 and g(0) = 1 (ii) f'(x) = g(x) (iii) g'(x) = f(x)
Q: How do we use the chain rule to differentiate the functions f(x) and g(x)?
A: We can use the chain rule to differentiate the functions f(x) and g(x) by writing:
f'(x) = g(x) g'(x) = f(x)
Using the chain rule, we can write:
f''(x) = g'(x) * g'(x) = f(x) * f(x) = (f(x))^2 g''(x) = f'(x) * f'(x) = g(x) * g(x) = (g(x))^2
Q: How do we solve the differential equations f''(x) = (f(x))^2 and g''(x) = (g(x))^2?
A: We can solve the differential equations f''(x) = (f(x))^2 and g''(x) = (g(x))^2 by using the following substitutions:
u = f(x) v = g(x)
Then, we have:
u' = f'(x) u'' = f''(x) v' = g'(x) v'' = g''(x)
Substituting u'' = (u)^2 and v'' = (v)^2, we get:
u'' = (u)^2 v'' = (v)^2
These are separable differential equations, which can be solved as follows:
∫(1/(u^2)) du = ∫dx -1/u = x + C
∫(1/(v^2)) dv = ∫dx -1/v = x + D
Q: How do we find the functions f(x) and g(x) using the initial conditions?
A: We can find the functions f(x) and g(x) using the initial conditions by substituting the values of C and D into the solutions to the differential equations.
From the solution to the differential equation for f(x), we have:
-1/u = x + C u = -1/(x + C)
Substituting u = f(x), we get:
f(x) = -1/(x + C)
Similarly, from the solution to the differential equation for g(x), we have:
-1/v = x + D v = -1/(x + D)
Substituting v = g(x), we get:
g(x) = -1/(x + D)
Q: What are the final answers for the functions f(x) and g(x)?
A: The final answers for the functions f(x) and g(x) are:
f(x) = -1/(x - 1/2) g(x) = -1/(x + 1)
Conclusion
In this Q&A article, we have provided additional insights and clarification on the solution to the problem of finding two interconnected functions f(x) and g(x) that satisfy specific conditions. We hope that this article has been helpful in understanding the solution and providing a clear and concise explanation of the steps involved.