Help pleasee 100 points

Mathematics

2 answers:

koban [17]3 years ago

7 0

rebrenta previous ly had 50 dollars

IrinaK [193]3 years ago

5 0

1. y=-4x+5
2. y=3x-8
3. Roberta had $50 before she started to save money each week.

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Giải phương trình y"= lnx

kipiarov [429]

Integrate both sides with respect to x :

$y''(x) = \ln(x) \implies \displaystyle \int y''(x)\,\mathrm dx = \int \ln(x)\,\mathrm dx$

On the right side, integrate by parts with

$f = \ln(x) \implies \mathrm df = \dfrac{\mathrm dx}x \\\\ \mathrm dg = \mathrm dx \implies g = x$

$\implies \displaystyle \int\ln(x)\,\mathrm dx = fg - \int g\,\mathrm df \\\\ \int\ln(x)\,\mathrm dx= x\ln(x) - \int \mathrm dx \\\\ \int\ln(x)\,\mathrm dx= x\ln(x) - x + C_1$

Then

$\displaystyle y'(x) = x\ln(x) - x + C_1$

Integrate both sides with respect to x by parts again, this time with

$f = \ln(x) \implies \mathrm df = \dfrac{\mathrm dx}x \\\\ \mathrm dg = x\,\mathrm dx \implies g = \dfrac{x^2}2$

$\implies \displaystyle \int x\ln(x)\,\mathrm dx = fg-\int g\,\mathrm df \\\\ \int x\ln(x)\,\mathrm dx = \dfrac{x^2\ln(x)}2 - \int\frac x2\,\mathrm dx \\\\ \int x\ln(x)\,\mathrm dx = \dfrac{x^2\ln(x)}2-\dfrac{x^2}4 + C_2$