how do you solve the initial value problem by using separation of variables dy/dx=1+x/(sqrt of y), y(2)=9
To solve the initial value problem dy/dx = 1 + x/(sqrt(y)), y(2) = 9 using separation of variables, we follow these steps:
Step 1: Rewrite the differential equation
The first step is to rewrite the equation in a form suitable for separation of variables. In this case, we have dy/dx = 1 + x/(sqrt(y)). We can rearrange it as:
1/(1 + x/(sqrt(y))) dy = dx
Step 2: Separate the variables
Next, we separate the variables by multiplying both sides of the equation by (1 + x/(sqrt(y))). This gives us:
dy = (1 + x/(sqrt(y))) dx
Step 3: Integrate both sides
Integrate both sides of the equation separately with respect to their respective variables. The integral of dy is simply y, and the integral of dx is x. This gives us:
y = ∫(1 + x/(sqrt(y))) dx
Step 4: Evaluate the integral
Now, we need to determine the integral of (1 + x/(sqrt(y))) with respect to x. The integral can be evaluated using techniques like substitution or integrating factors.
Step 5: Solve for y
After evaluating the integral, we have an equation in the form y = f(x), where f(x) is some function of x.
Step 6: Apply the initial condition
Finally, we apply the initial condition y(2) = 9 to determine the arbitrary constant and find the particular solution to the initial value problem.
It's important to note that the integration step may require advanced techniques depending on the complexity of the equation.
Watch missing parentheses, please.
Parentheses are needed to enclose numerators and denominators, otherwise additions and subtractions will take place after the division.
Separate the variables,
dy/dx = (1+x)/sqrt(y)
sqrt(y)dy = (1+x)dx
Integrate:
(2/3)y^(3/2) = x + x²/2 + C
y = [(3/2)(x+x²/2+C)]^(2/3)
from which we can solve for C=14.
so
y(x)=[(3/2)(x+x²/2+14)]^(2/3)
Please check all arithmetic.