Let f be the function satisfying f'(x)=x√(f(x)) for all real numbers x, where f(3)=25. 1. Find f''(3).
2. Write an expression for y=f(x) by solving the differential equation dy/dx=x√y with the initial condition f(3)=25.
Please show work and reasoning.
y' = x√y
y'(3) = 3√25 = 15
y'' = √y + x/(2√y) y'
y''(3) = 5 + 3/10 * 15 = 9.5
y' = x√y
dy/√y = x dx
2√y = 1/2 x^2 + c
y(3) = 25, so
10 = 9/2 + c
c = 11/2
4√y = x^2+c
y = (x^2+11)^2/16
check: y' = x(x^2+11)/4 = x√y
1. To find f''(3), we need to find the second derivative of f.
Given f'(x) = x√(f(x)), we can differentiate both sides of this equation with respect to x:
d/dx(f'(x)) = d/dx(x√(f(x)))
By applying the chain rule, we have:
f''(x) = 1√(f(x)) + x * (1/2) * (f(x))^(-1/2) * f'(x)
Substituting f'(x) = x√(f(x)), we get:
f''(x) = 1/√(f(x)) + x * (1/2) * (f(x))^(-1/2) * (x√(f(x)))
Now let's evaluate f''(3):
f''(3) = 1/√(f(3)) + 3 * (1/2) * (f(3))^(-1/2) * (3√(f(3)))
Since we know f(3) = 25, we substitute this value:
f''(3) = 1/√(25) + 3 * (1/2) * (25)^(-1/2) * (3√(25))
Simplifying further:
f''(3) = 1/5 + 3 * (1/2) * (25)^(-1/2) * 3
f''(3) = 1/5 + 3/2 * 5 * 3
f''(3) = 1/5 + 9
f''(3) = 1/5 + 45/5
f''(3) = 46/5
Therefore, f''(3) = 46/5.
2. To find the expression for y = f(x), we need to solve the differential equation dy/dx = x√y with the initial condition f(3) = 25.
Let's separate variables and integrate:
∫(1/√y) dy = ∫x dx
2√y = x^2/2 + C
Squaring both sides:
4y = x^2 + C'
Using the initial condition f(3) = 25, we substitute x = 3 and y = 25:
4(25) = 3^2 + C'
100 = 9 + C'
C' = 100 - 9 = 91
Therefore, the expression for y = f(x) is:
4y = x^2 + 91
y = (x^2 + 91)/4
To find the second derivative f''(x), we need to differentiate the given equation f'(x) = x√(f(x)) with respect to x.
1. Differentiating both sides of the equation with respect to x, we have:
f''(x) = (d/dx) (x√(f(x)))
Using the product rule and the chain rule, we get:
f''(x) = √(f(x)) + x*(1/2)*(1/√(f(x))) * f'(x)
Simplifying further, we get:
f''(x) = √(f(x)) + (x/2)*(1/√(f(x))) * x√(f(x))
f''(x) = √(f(x)) + (x^2)/(2√(f(x)))
To find f''(3), we substitute x = 3 into the equation:
f''(3) = √(f(3)) + (3^2)/(2√(f(3)))
Since f(3) = 25, we have:
f''(3) = √(25) + (9)/(2√(25))
f''(3) = 5 + 9/10
f''(3) = 5.9
Therefore, f''(3) = 5.9.
2. To solve the differential equation dy/dx = x√y with the initial condition f(3) = 25, we need to separate the variables and integrate.
dy/√y = xdx
Integrating both sides:
∫dy/√y = ∫xdx
On the left-hand side, we use the substitution u = √y, which gives du = (1/2)(1/√y)dy:
∫(1/u)du = ∫xdx
ln|u| = (1/2)x^2 + C1
Substituting back u = √y and simplifying:
ln|√y| = (1/2)x^2 + C1
Using the logarithmic property ln|√y| = (1/2)ln|y|:
ln|y| = 2[(1/2)x^2 + C1]
ln|y| = x^2 + C2
Taking the exponentiation of both sides:
|y| = e^(x^2 + C2)
Since y = f(x), we can write the equation as:
|f(x)| = e^(x^2 + C2)
Finally, using the initial condition f(3) = 25, we can find the value of C2:
|25| = e^(3^2 + C2)
25 = e^(9 + C2)
Taking the natural logarithm of both sides:
ln(25) = 9 + C2
C2 = ln(25) - 9
Therefore, the expression for y = f(x) is:
f(x) = ±e^(x^2 + ln(25) - 9)
This can be further simplified as:
f(x) = ±e^(x^2 - 9) * 25
To find f''(3), we need to differentiate f'(x) with respect to x.
Step 1: Differentiate f'(x) using the chain rule:
d/dx [ f'(x) ] = d/dx (x√f(x))
Step 2: Apply the product rule to the differentiated expression:
d/dx [ f'(x) ] = x * d/dx √f(x) + √f(x) * d/dx x
Step 3: Simplify by differentiating each term:
d/dx [ f'(x) ] = x * 1/(2√f(x)) * f'(x) + √f(x)
Step 4: Substitute the given equation f'(x) = x√f(x):
d/dx [ f'(x) ] = x * 1/(2√f(x)) * (x√f(x)) + √f(x)
Step 5: Simplify further:
d/dx [ f'(x) ] = x^2/(2√f(x)) + √f(x)
Now, we can find f''(3) by evaluating this expression at x = 3:
f''(3) = (3^2)/(2√f(3)) + √f(3)
Given that f(3) = 25, we can substitute this value into the equation:
f''(3) = (3^2)/(2√25) + √25
=> f''(3) = (9/10) + 5
=> f''(3) = 9/10 + 50/10
=> f''(3) = 59/10
Therefore, f''(3) = 59/10.
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To write the expression for y = f(x) by solving the differential equation dy/dx = x√y with the initial condition f(3) = 25, we can use the method of separation of variables.
Step 1: Begin by separating the variables:
dy/√y = x dx
Step 2: Integrate both sides:
∫dy/√y = ∫x dx
Step 3: Evaluate the integrals:
2√y = (1/2)x^2 + C
Step 4: Solve for y:
√y = (1/4)x^2 + C/2
Step 5: Remove the square root by squaring both sides:
y = [(1/4)x^2 + C/2]^2
Step 6: Use the initial condition f(3) = 25 to find the value of the constant C:
25 = [(1/4)(3)^2 + C/2]^2
25 = [(9/4) + C/2]^2
5 = (9/4) + C/2
5 - 9/4 = C/2
20/4 - 9/4 = C/2
11/4 = C/2
C = 11/2
Step 7: Substitute the value of C back into the equation:
y = [(1/4)x^2 + 11/4]^2
Therefore, the expression for y = f(x) by solving the differential equation with the initial condition f(3) = 25 is:
y = [(1/4)x^2 + 11/4]^2.