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Answer :
To solve the problem, we need to find the correct function [tex]\( y \)[/tex] that satisfies the given differential equation [tex]\( y' = -4 \sqrt{x} + 24 e^{-6x} - 9 \sin x \)[/tex] and also passes through the point [tex]\((0, 8)\)[/tex].
Here are the steps to identify the correct function:
1. Analyze Candidate Solutions:
We have four candidate solutions provided:
- [tex]\( y_1 = -\frac{8}{3} x^{\frac{3}{2}} - 4 e^{-6x} + 9 \cos x + 3 \)[/tex]
- [tex]\( y_2 = -6 x^{\frac{3}{2}} - 4 e^{-6x} - 9 \cos x + 21 \)[/tex]
- [tex]\( y_3 = -6 x^{\frac{3}{2}} - 4 e^{-6x} + 9 \cos x + 3 \)[/tex]
- [tex]\( y_4 = -\frac{8}{3} x^{\frac{3}{2}} - 4 e^{-6x} - 9 \cos x + 21 \)[/tex]
2. Check Passing Through the Point (0, 8):
For each candidate solution, substitute [tex]\( x = 0 \)[/tex] and see if [tex]\( y = 8 \)[/tex].
- For [tex]\( y_1 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\( y_1 = -\frac{8}{3} \cdot 0^{\frac{3}{2}} - 4 e^{-6 \cdot 0} + 9 \cos(0) + 3 \)[/tex]
Simplifying gives:
[tex]\( y_1 = 0 - 4 \cdot 1 + 9 \cdot 1 + 3 = 0 - 4 + 9 + 3 = 8 \)[/tex]
Since this satisfies the point [tex]\((0, 8)\)[/tex], [tex]\( y_1 \)[/tex] could be the solution.
3. Verify Differential Equation:
Finally, we need to check if the derivative of this [tex]\( y_1 \)[/tex], denoted as [tex]\( y_1' \)[/tex], matches the given differential equation:
- Calculate [tex]\( y_1' \)[/tex]:
- The derivative of [tex]\(-\frac{8}{3} x^{\frac{3}{2}}\)[/tex] is [tex]\(-4\sqrt{x}\)[/tex].
- The derivative of [tex]\(-4 e^{-6x}\)[/tex] is [tex]\(24 e^{-6x}\)[/tex].
- The derivative of [tex]\(9 \cos x\)[/tex] is [tex]\(-9 \sin x\)[/tex].
- The derivative of the constant 3 is 0.
- Thus, [tex]\( y_1' = -4 \sqrt{x} + 24 e^{-6x} - 9 \sin x \)[/tex], which matches the given differential equation.
Since [tex]\( y_1 \)[/tex] passes through the point [tex]\((0, 8)\)[/tex] and its derivative matches the given expression for [tex]\( y'\)[/tex], the correct solution is:
[tex]\( y = -\frac{8}{3} x^{\frac{3}{2}} - 4 e^{-6x} + 9 \cos x + 3 \)[/tex].
Here are the steps to identify the correct function:
1. Analyze Candidate Solutions:
We have four candidate solutions provided:
- [tex]\( y_1 = -\frac{8}{3} x^{\frac{3}{2}} - 4 e^{-6x} + 9 \cos x + 3 \)[/tex]
- [tex]\( y_2 = -6 x^{\frac{3}{2}} - 4 e^{-6x} - 9 \cos x + 21 \)[/tex]
- [tex]\( y_3 = -6 x^{\frac{3}{2}} - 4 e^{-6x} + 9 \cos x + 3 \)[/tex]
- [tex]\( y_4 = -\frac{8}{3} x^{\frac{3}{2}} - 4 e^{-6x} - 9 \cos x + 21 \)[/tex]
2. Check Passing Through the Point (0, 8):
For each candidate solution, substitute [tex]\( x = 0 \)[/tex] and see if [tex]\( y = 8 \)[/tex].
- For [tex]\( y_1 \)[/tex], substitute [tex]\( x = 0 \)[/tex]:
[tex]\( y_1 = -\frac{8}{3} \cdot 0^{\frac{3}{2}} - 4 e^{-6 \cdot 0} + 9 \cos(0) + 3 \)[/tex]
Simplifying gives:
[tex]\( y_1 = 0 - 4 \cdot 1 + 9 \cdot 1 + 3 = 0 - 4 + 9 + 3 = 8 \)[/tex]
Since this satisfies the point [tex]\((0, 8)\)[/tex], [tex]\( y_1 \)[/tex] could be the solution.
3. Verify Differential Equation:
Finally, we need to check if the derivative of this [tex]\( y_1 \)[/tex], denoted as [tex]\( y_1' \)[/tex], matches the given differential equation:
- Calculate [tex]\( y_1' \)[/tex]:
- The derivative of [tex]\(-\frac{8}{3} x^{\frac{3}{2}}\)[/tex] is [tex]\(-4\sqrt{x}\)[/tex].
- The derivative of [tex]\(-4 e^{-6x}\)[/tex] is [tex]\(24 e^{-6x}\)[/tex].
- The derivative of [tex]\(9 \cos x\)[/tex] is [tex]\(-9 \sin x\)[/tex].
- The derivative of the constant 3 is 0.
- Thus, [tex]\( y_1' = -4 \sqrt{x} + 24 e^{-6x} - 9 \sin x \)[/tex], which matches the given differential equation.
Since [tex]\( y_1 \)[/tex] passes through the point [tex]\((0, 8)\)[/tex] and its derivative matches the given expression for [tex]\( y'\)[/tex], the correct solution is:
[tex]\( y = -\frac{8}{3} x^{\frac{3}{2}} - 4 e^{-6x} + 9 \cos x + 3 \)[/tex].
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