Which of the following is the general solution of the differential equation dy/dx = 2x/y?
y2 = x2 + C
y2 = 2x2 + C <- my answer
y2 = 4x2 + C
x2 − y2 = C
yes.
To find the general solution of the given differential equation, dy/dx = 2x/y, we can follow these steps:
Step 1: Rearrange the equation to separate variables:
y * dy = 2x * dx
Step 2: Integrate both sides with respect to their respective variables:
∫y * dy = ∫2x * dx
Step 3: Evaluate the integrals:
y^2/2 = x^2 + C, where C is the constant of integration.
Step 4: Multiply both sides by 2 to eliminate the fraction:
y^2 = 2x^2 + 2C
Therefore, the correct general solution of the differential equation is y^2 = 2x^2 + 2C.
To find the general solution of the given differential equation dy/dx = 2x/y, we can use separation of variables method. Here's how you can do it:
1. Start by rearranging the equation to separate variables:
dy/y = 2x dx
2. Integrate both sides of the equation with respect to their respective variables:
∫(dy/y) = ∫(2x dx)
3. Integrating the left side gives us:
ln|y| = x^2 + C1, where C1 is the constant of integration.
4. Solve for y by exponentiating both sides:
|y| = e^(x^2 + C1)
5. Remove the absolute value by considering both positive and negative cases:
y = ± e^(x^2 + C1)
6. Combine the constants into a single constant C:
y = ± e^(x^2) * e^(C)
7. Simplify the constant to a new constant C:
y = ± K * e^(x^2), where K = e^C is a new constant.
Therefore, the general solution of the differential equation dy/dx = 2x/y is y = ± K * e^(x^2), where K is a constant. None of the given options match the general solution.