Find the domain for the particular solution to the differential equation dy dx equals the quotient of 3 times y and x, with initial condition y(1) = 1.
To find the domain for the particular solution to the given differential equation, we first need to analyze the equation itself.
The differential equation is given as:
dy/dx = (3y) / x
From the equation, we can observe that the term "x" is in the denominator. Therefore, the domain of "x" should not include any value that makes the denominator equal to zero.
To determine the restricted values for "x," we set the denominator equal to zero and solve:
x = 0
Hence, "x = 0" is not in the domain of the solution.
Now, let's consider the initial condition given: y(1) = 1
Since the initial condition specifies a value of "y" at "x = 1," we know that the solution exists at least for "x = 1." Therefore, the domain of the solution will include "x = 1."
In conclusion, the domain of the particular solution to the given differential equation is all real numbers except 0.
To find the domain for the particular solution to the differential equation dy/dx = (3y)/x with the initial condition y(1) = 1, we can first analyze the equation itself.
The equation dy/dx = (3y)/x represents a separable first-order ordinary differential equation. We can solve it using separation of variables.
Step 1: Separate the variables
dy/y = (3/x)dx
Step 2: Integrate both sides
∫(dy/y) = ∫(3/x)dx
ln|y| = 3ln|x| + C1
Step 3: Solve for y
Taking the exponential of both sides, we get:
|y| = e^(3ln|x| + C1)
Since e^(3ln|x| + C1) = e^(ln|x|^3) * e^(C1) = (e^(ln|x|^3)) * e^(C1) = |x|^3 * e^(C1), we can rewrite the equation as:
|y| = |x|^3 * e^(C1)
Step 4: Consider the initial condition
Since the initial condition is y(1) = 1, we can substitute x = 1 and y = 1 into the equation:
|1| = |1|^3 * e^(C1)
1 = 1 * e^(C1)
1 = e^(C1)
From here, we see that C1 must be 0, since e^0 = 1.
Therefore, our equation becomes:
|y| = |x|^3
To find the domain, we need to determine the values of x for which the absolute value of y is defined.
For this equation, y can be any real number, but x must be greater than or equal to 0 since we have |x|^3.
So, the domain for the particular solution to the differential equation is x ≥ 0.
why all those useless words? Use some math symbols!
dy/dx = 3y/x
dy/y = 3/x dx
lny = 3lnx + c
or,
lny = ln(x^3) + ln(c) (different c)
y = cx^3
y(1)=1 means c=1
y = x^3
I expect that you can determine the domain for that...