Certainly! Let's work through the given differential equation step-by-step.
We are given:
[tex]\[ 5 \sin x + 7 \cos y = 2 \][/tex]
[tex]\[ \frac{d y}{d x} = 5 \][/tex]
### Step 1: Understand the Given Equations
- The first equation is a relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- The second is a first-order differential equation indicating how [tex]\(y\)[/tex] changes with respect to [tex]\(x\)[/tex].
### Step 2: Differentiating the Implicit Function
To analyze the given relationship, let’s differentiate the equation [tex]\(5 \sin x + 7 \cos y = 2\)[/tex] with respect to [tex]\(x\)[/tex].
Implicit differentiation:
[tex]\[ \frac{d}{dx} \left(5 \sin x + 7 \cos y \right) = \frac{d}{dx}(2)\][/tex]
Since 2 is a constant, its derivative with respect to [tex]\(x\)[/tex] is 0:
[tex]\[ 5 \frac{d}{dx} (\sin x) + 7 \frac{d}{dx} (\cos y) = 0 \][/tex]
Now, apply the chain rule where necessary:
[tex]\[ 5 \cos x + 7 \left(-\sin y \frac{d y}{d x} \right) = 0 \][/tex]
Substitute [tex]\(\frac{d y}{d x} = 5\)[/tex]:
[tex]\[ 5 \cos x - 7 \sin y (5) = 0 \][/tex]
This simplifies to:
[tex]\[ 5 \cos x - 35 \sin y = 0 \][/tex]
### Step 3: Solving for [tex]\(\cos x\)[/tex]
[tex]\[ 5 \cos x = 35 \sin y \][/tex]
Divide each side by 5:
[tex]\[ \cos x = 7 \sin y \][/tex]
### Conclusion:
Based on this derived relationship:
[tex]\[ \cos x = 7 \sin y \][/tex]
This relationship is necessary for ensuring that the given differential equation [tex]\(\frac{d y}{d x} = 5\)[/tex] holds true for the original function [tex]\(5 \sin x + 7 \cos y = 2\)[/tex].
Thus, by handling implicit differentiation and respecting the given conditions, we conclude that for the given equation [tex]\(5 \sin x + 7 \cos y = 2\)[/tex], the differential [tex]\(\frac{d y}{d x} = 5\)[/tex] is consistent with the relationship [tex]\(\cos x = 7 \sin y\)[/tex].
The result is consistent with the correct analysis of the given problem.
