Appearance
question:Solve the initial-value problem: {eq}; x{y}' + 2y = 3x^2 , , ; y(3) = 5 {/eq}.
answer:Given the differential equation: {eq}displaystyle x{y}' + 2y = 3x^2 {/eq} Multiply through by {eq}x {/eq} to obtain: {eq}displaystyle x^2{y}' + 2xy = 3x^3 {/eq} This simplifies to the derivative of {eq}x^2y {/eq}: {eq}displaystyle (x^2y)' = 3x^3 {/eq} Integrate both sides: {eq}Rightarrow displaystyle x^2y = int 3x^3 dx = 3dfrac{x^4}{4} + C {/eq} Divide by {eq}x^2 {/eq} to find {eq}y {/eq}: {eq}Rightarrow displaystyle y = dfrac{3x^4}{4x^2} + dfrac{C}{x^2} = dfrac{3x^2 + 4C}{4x^2} {/eq} Using the initial condition {eq}y(3) = 5 {/eq}, we have: {eq}5 = dfrac{3(3)^2 + 4C}{4(3)^2} Rightarrow C = dfrac{3(3)^2 - 4 cdot 5 cdot 3^2}{4} = -dfrac{63}{4} {/eq} Thus, the solution is: {eq}displaystyle y = dfrac{3x^2 - 63}{4x^2} {/eq} The final solution to the initial-value problem is: {eq}displaystyle y = dfrac{3x^2 - 63}{4x^2} {/eq}
question:What poetic structures are employed in Proverbs 31?
answer:Proverbs 31 incorporates two poetic forms: it is both an acrostic poem and a chiasm. An acrostic poem is evident in verses 10 through 31, where each line's initial Hebrew letter sequentially follows the alphabet, highlighting the virtues of an ideal wife. A chiasm, on the other hand, is a type of Hebrew poetry characterized by parallel lines with a central focus. In Proverbs 31, this structure is observed as the lines pair in descending and ascending order, with verse 23 serving as the central emphasis. This middle line describes the attributes of a commendable Christian man and husband. The chiastic pattern underscores the symmetry and balance in the poem's presentation of these virtues.
question:Record the journal entries for the disposal of equipment with a book value of 184,250, which was originally acquired for 273,500 and has accumulated depreciation of 89,250. 1. If the equipment is sold for 170,500. 2. If the equipment is sold for 189,000.
answer:1. For the equipment sold at 170,500 (below book value), there is a loss: {eq}Loss = Book:value - Selling:price = 184,250 - 170,500 = 13,750 {/eq} Journal entry: | Debit | Credit | | --- | --- | | Cash | 170,500 | | Accumulated Depreciation | 89,250 | | Loss on Sale of Equipment | 13,750 | | Equipment | 273,500 | 2. For the equipment sold at 189,000 (above book value), there is a gain: {eq}Gain = Selling:price - Book:value = 189,000 - 184,250 = 4,750 {/eq} Journal entry: | Debit | Credit | | --- | --- | | Cash | 189,000 | | Accumulated Depreciation | 89,250 | | Equipment | 273,500 | | Gain on Sale of Equipment | 4,750 |
question:Find all real solutions to the equation sqrt{frac{7}{2}-9 x}+sqrt{-5 x-frac{15}{2}}=8.
answer:The square root of a negative number is not a real number. Therefore, the equation sqrt{-5 x-frac{15}{2}}=8 has no real solutions. Since the sum of two non-real numbers cannot equal a real number, the equation sqrt{frac{7}{2}-9 x}+sqrt{-5 x-frac{15}{2}}=8 also has no real solutions. The equation sqrt{frac{7}{2}-9 x}+sqrt{-5 x-frac{15}{2}}=8 has no real solutions.