The funtions F and G are defined as follows
1 answer:
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Answer:
see explanation
Step-by-step explanation:
The bar above the 3 indicates the 3 is being repeated, that is
0.13333....
We require to make 2 equations with the repeating 3 after the decimal point.
let x = 0.13333... , then multiply both sides by 10 then 100
10x = 1.3333 ← equation with repeating 3 after the decimal point
100x = 13.3333 ← second equation with repeated 3 after the decimal point
Subtracting the equations eliminates the repeated 3, that is
100x - 10x = 13.333 - 1.333
90x = 12 ( divide both sides by 90 )
x = ← divide numerator/denominator by 6, hence
0.1333... =
Answer:
y = x⁴ + x³ - 3x² + 5x + C
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Separable differential equations such as these ones can be solved by treating dy/dx as a ratio of differentials. Then move the dx with all the x terms and move the dy with all the y terms. After that, integrate both sides of the equation.
In general (understood that +C portions are still there),
Note that ∫dy = y since it is ∫1·dy = ∫y⁰ dy = y¹/(0+1) = y
For the right-hand side, we use the sum/difference rule for integrals, which says that
![\int \big[f(x) \pm g(x)\big]\, dx = \int f(x)\,dx \pm \int g(x) \, dx](https://tex.z-dn.net/?f=%5Cint%20%5Cbig%5Bf%28x%29%20%5Cpm%20g%28x%29%5Cbig%5D%5C%2C%20dx%20%3D%20%5Cint%20f%28x%29%5C%2Cdx%20%5Cpm%20%5Cint%20g%28x%29%20%5C%2C%20dx)
Applying these concepts:
The answer is y = x⁴ + x³ - 3x² + 5x + C
y = x⁴ + x³ - 3x² + 5x + C
======
Separable differential equations such as these ones can be solved by treating dy/dx as a ratio of differentials. Then move the dx with all the x terms and move the dy with all the y terms. After that, integrate both sides of the equation.
In general (understood that +C portions are still there),
Note that ∫dy = y since it is ∫1·dy = ∫y⁰ dy = y¹/(0+1) = y
For the right-hand side, we use the sum/difference rule for integrals, which says that
Applying these concepts:
The answer is y = x⁴ + x³ - 3x² + 5x + C