Find the particular solution (solved for y) for the differential equation dy/dx=2x/e^(2y) satisfying y(0)=1.
e^-2y dy = 2 x dx
e-2y /-2 = x^2 + c
e^-2y = -2 x^2 + c
-2y = ln (-2 x^2+c)
when x = 0, y = 1
-2 = ln(c)
e^-2 = c
-2 y = ln (-2x^2+e^-2)
y = -(1/2) ln(-2x^2+1/e^2)
Can you explain why it's simplified to e^-2ydy=2xdx. I don't understand why it's e^negative 2y
To find the particular solution of the differential equation dy/dx = 2x/e^(2y) satisfying y(0) = 1, we can use the method of separation of variables.
Step 1: Write the given differential equation in the form of dy/dx = f(x)/g(y):
dy/dx = 2x/e^(2y)
Step 2: Separate the variables by multiplying both sides of the equation by e^(2y):
e^(2y)dy = 2x dx
Step 3: Integrate both sides of the equation with respect to their respective variables:
∫e^(2y)dy = ∫2x dx
Step 4: Integrate each side of the equation:
∫e^(2y)dy = 2∫xdx
Integrating the left side:
∫e^(2y)dy = (∫1/2)e^(2y)dy = 1/2 * e^(2y) + C1
Integrating the right side:
2∫xdx = 2 * (1/2)x^2 + C2 = x^2 + C2
Step 5: Set up the equation using the initial condition y(0) = 1:
1/2 * e^(2y) + C1 = x^2 + C2
Substituting x = 0 and y = 1:
1/2 * e^(2*1) + C1 = 0^2 + C2
1/2 * e^2 + C1 = C2
Step 6: Simplify the equation:
1/2 * e^2 + C1 = C2
Step 7: Rearrange the equation to solve for e^2:
e^2 = 2(C2 - C1)
Step 8: Substitute e^2 back into the equation:
1/2 * e^(2y) + C1 = x^2 + 2(C2 - C1)
Step 9: Simplify the equation:
1/2 * e^(2y) = x^2 + 2(C2 - C1) - C1
1/2 * e^(2y) = x^2 + C2 - 2C1
Step 10: Solve for y:
e^(2y) = 2(x^2 + C2 - 2C1)
2y = ln(2(x^2 + C2 - 2C1))
y = (1/2) * ln(2(x^2 + C2 - 2C1)) + C3
Thus, the particular solution for the given differential equation dy/dx = 2x/e^(2y) satisfying y(0) = 1 is y = (1/2) * ln(2(x^2 + C2 - 2C1)) + C3.
To find the particular solution for the given differential equation, we can separate the variables and then integrate both sides.
The given differential equation is:
dy/dx = 2x/e^(2y)
Let's start by separating the variables. We can bring all the terms involving y to one side and all the terms involving x to the other side:
e^(2y)dy = 2x dx
Now, we can integrate both sides.
∫e^(2y)dy = ∫2x dx
Integrating the left side with respect to y:
∫e^(2y)dy = ∫2x dx
By using the substitution u = 2y, we can simplify the left side:
(1/2) ∫e^u du = ∫2x dx
Integrating both sides will give us:
(1/2) * e^u = x^2 + C
Now substitute back u = 2y:
(1/2) * e^(2y) = x^2 + C
Now, to find the particular solution satisfying y(0) = 1, we substitute x = 0 and y = 1 into the equation:
(1/2) * e^(2(1)) = 0^2 + C
(1/2) * e^2 = C
Therefore, the value of C is (1/2) * e^2.
Substituting this value back into the equation gives us the particular solution solved for y:
(1/2) * e^(2y) = x^2 + (1/2) * e^2
So, the particular solution for the given differential equation, solved for y, is:
y(x) = (1/2) * ln(x^2 + (1/2) * e^2)