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muminat
2 years ago
7

Giải phương trình y"= lnx

Mathematics
1 answer:
kipiarov [429]2 years ago
3 0

Integrate both sides with respect to <em>x</em> :

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 <em>x</em> 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

Then

y(x) = \dfrac{x^2\ln(x)}2 - \dfrac{x^2}4 - \dfrac{x^2}2 + C_1x + C_2 \\\\ \boxed{y(x) = \dfrac{(2\ln(x)-3)x^2}4 + C_1x + C_2}

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Ok this is a very question that can solved very easily. So we know that there three numbers, and they are consecutive, this means that they are all one bigger than the number before so for example 1,2, and 3 or 20, 21, and 22. These numbers are called consecutive. Now let's just pretend that the answer to this question is the variable "x". Now x is the smallest number in all the three numbers.

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