hi i need to know how to do this one
thanks
Find the solution to yx + x + (dy/dx) = 0 ; y(9) = -9
y = ????
Does this separate the variables?
dy/dx=-x(y+1)
dy/(y+1) = -x dx
To find the solution to the given differential equation, you can use the method of separating variables. Here's how to do it:
1. Start with the differential equation:
dy/dx = -x(y + 1)
2. Rearrange the equation to separate the variables:
dy / (y + 1) = -x dx
3. Integrate both sides of the equation with respect to their respective variables:
∫ dy / (y + 1) = -∫ x dx
4. Evaluate the integrals:
ln|y + 1| = -0.5x^2 + C1
Here, C1 is the constant of integration.
5. Solve for y:
Take the exponential of both sides:
e^(ln|y + 1|) = e^(-0.5x^2 + C1)
Simplify:
|y + 1| = e^(C1) * e^(-0.5x^2)
Since e^(C1) is just another constant, we'll replace it with C2:
|y + 1| = C2 * e^(-0.5x^2)
6. Consider the absolute value:
Since the absolute value of y + 1 can be positive or negative, we need to consider two cases:
Case 1: (y + 1) > 0
In this case, we can drop the absolute value:
y + 1 = C2 * e^(-0.5x^2)
Rearrange to solve for y:
y = C2 * e^(-0.5x^2) - 1
Case 2: (y + 1) < 0
In this case, we need to change the sign and remove the absolute value:
-(y + 1) = C2 * e^(-0.5x^2)
Rearrange to solve for y:
y = -1 - C2 * e^(-0.5x^2)
7. Apply the initial condition:
We are given y(9) = -9.
Substituting this value into the equation, we get:
-9 = C2 * e^(-0.5(9)^2) - 1
Solve for C2:
C2 * e^(-0.5(9)^2) = -9 + 1
C2 * e^(-0.5(9)^2) = -8
C2 = -8 / e^(-0.5(9)^2)
8. Substitute the value of C2 back into the solutions obtained in step 6 for each case to obtain the final solutions for y.
I hope this explanation helps you solve the differential equation and find the solution for y.