Solve The Separable Differential Equation Dx/Dt = X^2 + 1/49, And Find The Particular Solution Satisfying The Initial Condition X(0) = -2.
The Correct Answer and Explanation is :
It seems that the image you provided isn’t accessible in this environment, but based on your description of the separable differential equation:
[
\frac{dx}{dt} = x^2 + \frac{1}{49}
]
Let’s proceed to solve it step by step and find the particular solution for the initial condition ( x(0) = -2 ).
Step 1: Separate the variables
We begin by separating the variables (x) and (t). To do this, we rewrite the equation as:
[
\frac{dx}{x^2 + \frac{1}{49}} = dt
]
Step 2: Simplify the left-hand side
The left-hand side involves a rational function in terms of (x). First, we factor out ( \frac{1}{49} ) from the denominator:
[
x^2 + \frac{1}{49} = \frac{49x^2 + 1}{49}
]
So the equation becomes:
[
\frac{dx}{\frac{49x^2 + 1}{49}} = dt
]
Simplifying further:
[
\frac{49dx}{49x^2 + 1} = dt
]
Step 3: Integrate both sides
Now, we integrate both sides. The integral on the left-hand side is a standard form that can be handled using a standard trigonometric substitution. The integral is:
[
\int \frac{49}{49x^2 + 1} dx = \int dt
]
This is a standard integral form where:
[
\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right)
]
In our case, ( a = \frac{1}{7} ), so:
[
\int \frac{49}{49x^2 + 1} dx = 7 \arctan(7x)
]
Thus, we now have:
[
7 \arctan(7x) = t + C
]
Step 4: Solve for the constant of integration
Now, we use the initial condition ( x(0) = -2 ) to find the value of the constant ( C ). Substituting ( t = 0 ) and ( x = -2 ) into the equation:
[
7 \arctan(7(-2)) = 0 + C
]
Simplifying:
[
7 \arctan(-14) = C
]
So, ( C = 7 \arctan(-14) ).
Final solution
Thus, the general solution is:
[
7 \arctan(7x) = t + 7 \arctan(-14)
]
To express ( x ) explicitly in terms of ( t ), we solve for ( x ):
[
\arctan(7x) = \frac{t}{7} + \arctan(-14)
]
[
7x = \tan\left(\frac{t}{7} + \arctan(-14)\right)
]
[
x = \frac{1}{7} \tan\left(\frac{t}{7} + \arctan(-14)\right)
]
This is the particular solution satisfying the initial condition ( x(0) = -2 ).
Conclusion:
The solution to the given differential equation, satisfying the initial condition, is:
[
x(t) = \frac{1}{7} \tan\left(\frac{t}{7} + \arctan(-14)\right)
]
This solution describes how ( x ) evolves over time based on the given equation.