Consider the differential equation given by dy/dx = xy/2.
A. Let y=f(x) be the particular solution to the given differential equation with the initial condition. Based on the slope field, how does the value of f(0.2) compare to f(0)? Justify your answer.
B. Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 3. Use your solution to find f(0.2).
dy/dx = xy/2
dy/y = x/2 dx
ln y = 1/4 x^2 + c
y = c e^(x^2/4)
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To solve the given differential equation, dy/dx = xy/2, we can separate the variables and integrate.
A. By visualizing the slope field, we can get an idea of how the function f(x) behaves. The slope field provides information about the slopes of different solutions at different points.
Since the slope in the slope field is directly proportional to the value of y, we can conclude that f(x) will be increasing as x increases. This means that the value of f(0.2) should be greater than f(0).
Justification: The slope field indicates that for each value of x, the slope of f(x) is positive. This implies that the function f(x) is increasing for positive values of x.
B. To find the particular solution y = f(x) with the initial condition f(0) = 3, we can follow these steps:
1. Separate the variables by multiplying both sides of the differential equation by 2/y:
2 dy/y = x dx
2. Integrate both sides with respect to their respective variables:
∫ 2/y dy = ∫ x dx
2 ln|y| = 1/2 x^2 + C
3. Solve for y by taking the exponentiation of both sides:
e^(2 ln|y|) = e^(1/2 x^2 + C)
y^2 = e^(1/2 x^2 + C)
y^2 = Ce^(1/2 x^2)
4. Apply the initial condition f(0) = 3 to find the value of C:
(3)^2 = C e^(1/2 (0)^2)
9 = C e^0
C = 9
5. Substitute back the value of C into the equation to obtain the particular solution:
y^2 = 9e^(1/2 x^2)
Taking the square root:
y = ± 3√(e^(1/2 x^2))
Finally, we can use the particular solution y = f(x) with the initial condition f(0) = 3 to find f(0.2):
f(0.2) = ± 3√(e^(1/2 (0.2)^2))
Calculating this expression will give you the value of f(0.2) depending on whether you take the positive or negative square root.
A. To determine how the value of f(0.2) compares to f(0) based on the slope field of the differential equation dy/dx = xy/2, we can follow these steps:
1. Draw a slope field: Draw short line segments at various points on a graph to represent the slope of the solution curve at those points. The slope at a given point (x, y) is determined by the equation dy/dx = xy/2.
2. Determine the behavior of the slope field: Observe the slope field to see how the slopes change with respect to the x and y coordinates. In this case, the slope is proportional to xy, which means that it depends on both x and y. The slope is positive when both x and y have the same sign, and negative when they have opposite signs.
3. Analyze the initial condition: The initial condition for this problem is not provided, so we cannot determine the exact value of f(0). However, we can make a logical assumption based on the behavior of the slope field.
4. Compare f(0) to f(0.2): Since the slopes in the slope field depend on both x and y, the exact values of f(0) and f(0.2) cannot be determined without further information. However, we can make a general comparison based on the slope field: if f(0) is positive, f(0.2) will also be positive, and if f(0) is negative, f(0.2) will also be negative. This is because the slopes in the slope field change gradually as we move along the curve.
Therefore, based on the slope field, we can conclude that the value of f(0.2) will have the same sign as f(0), but we cannot determine the exact relationship between the two values without additional information.
B. To find the particular solution y = f(x) to the given differential equation dy/dx = xy/2 with the initial condition f(0) = 3, we can use the method of separation of variables. Follow these steps:
1. Write the differential equation: dy/dx = xy/2.
2. Move variables to separate sides: Divide both sides by y/2 and dx to get dy/y = x dx/2.
3. Integrate both sides: ∫dy/y = ∫x dx/2.
The left side can be integrated as ln|y|, and the right side can be integrated as x^2/4 + C, where C is the constant of integration.
4. Solve for y: Now, we have ln|y| = x^2/4 + C. To eliminate the absolute value, we can raise both sides to the power of e:
e^(ln|y|) = e^(x^2/4 + C).
This simplifies to |y| = e^(x^2/4) * e^C.
Since e^C is always positive, we can remove the absolute value and write the equation as y = ±e^(x^2/4) * e^C.
5. Apply the initial condition: We are given that f(0) = 3. Substituting x = 0 in the equation, we get:
f(0) = ±e^(0^2/4) * e^C = ±e^0 * e^C = ±1 * e^C.
So, ±e^C = 3.
Since e^C is always positive, we must choose the positive sign, giving us e^C = 3.
Taking the natural logarithm of both sides, we have C = ln(3).
6. Write the particular solution: Now we have y = e^(x^2/4) * e^C, which simplifies to:
y = e^(x^2/4) * e^(ln(3)).
Using the property e^(a+b) = e^a * e^b, we can simplify this further:
y = e^(x^2/4 + ln(3)).
This is the particular solution to the given differential equation with the initial condition f(0) = 3.
7. Find f(0.2): To find f(0.2), substitute x = 0.2 into the particular solution equation:
f(0.2) = e^((0.2)^2/4 + ln(3)).
Use a calculator to evaluate the expression to find the value of f(0.2).
By following these steps, we can find the particular solution to the given differential equation with the initial condition, and use it to find the value of f(0.2).