Find the solution of the differential equation that satisfies the given initial condition.
dy/dx= x/y, y(0) = −7
Stave do you mean that the answer would be y=-sqrt(x^2+49)?
dy/dx = x/y
y dy = x dx
y^2 = x^2 + c
49 = 0 + c
c = 49
y^2 - x^2 = 49
yes.
Well, this differential equation looks like a real "y-killer"! But let's see if we can find a solution to satisfy that initial condition.
To tackle this, let's separate the variables to get rid of the y's hanging around. So we'll rewrite the equation as:
y dy = x dx
Now, let's integrate both sides. On the left side, we'll integrate with respect to y, and on the right side with respect to x. So we have:
∫ y dy = ∫ x dx
Integrating, we get:
(y^2)/2 = (x^2)/2 + C
Here, C is a constant of integration. Now, let's solve for y:
y^2 = x^2 + 2C
To satisfy the initial condition y(0) = -7, we can substitute x = 0 and y = -7 into the equation:
(-7)^2 = 0^2 + 2C
49 = 2C
C = 49/2
Now, let's substitute C back into our equation:
y^2 = x^2 + 49
Taking the square root of both sides gives us:
y = ±√(x^2 + 49)
So, the solution to the differential equation that satisfies the initial condition y(0) = -7 is:
y = ±√(x^2 + 49)
To solve the given differential equation, we can separate the variables and integrate both sides. Let's go through each step:
Step 1: Separate the variables:
dy/dx = x/y
Rearrange the equation to isolate dy on one side and dx on the other:
y dy = x dx
Step 2: Integrate both sides:
∫y dy = ∫x dx
Integrating the left side with respect to y and the right side with respect to x, we get:
(1/2) y^2 + C1 = (1/2) x^2 + C2
Here, C1 and C2 are constants of integration.
Step 3: Solve for y:
(1/2) y^2 = (1/2) x^2 + (C2 - C1)
Multiply both sides by 2 to get rid of the coefficient (1/2):
y^2 = x^2 + 2(C2 - C1)
Step 4: Apply the initial condition:
y(0) = -7
Substitute x = 0 and y = -7 into the equation:
(-7)^2 = 0^2 + 2(C2 - C1)
Simplifying:
49 = 2(C2 - C1)
Step 5: Solve for the constants:
Since C2 - C1 is a constant, we can rewrite 2(C2 - C1) as a single constant, say C3:
49 = 2(C2 - C1) = 2C3
Dividing both sides by 2:
C3 = 49/2 = 24.5
Step 6: Substitute the value of C3 back into the equation:
y^2 = x^2 + 2C3
Therefore, the solution to the differential equation that satisfies the initial condition y(0) = -7 is:
y^2 = x^2 + 2(49/2)
Simplifying further:
y^2 = x^2 + 49
Taking the square root of both sides:
y = ±√(x^2 + 49)
So the general solution is:
y = ±√(x^2 + 49)