Question
Download Solution PDFIf \(\frac{dy}{dx} = (\ln 5)y \) with y(0) = ln 5, then what is y(1) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFormula used:
- \(\int \frac{dy}{y} = ln \ y + C\)
- log m + log n = log mn
- log mn = n log m
Calculation:
Given that, \(\frac{dy}{dx} = (\ln 5)y \)
⇒ \(\frac{dy}{y} = (\ln 5)dx\)
Doing integration on both sides
⇒ \(\int\frac{dy}{y} =\int (\ln 5)dx\)
Since, \(\int \frac{dy}{y} = ln \ y + C\)
⇒ ln y = x ln 5 + ln C
Using the logarithmic property (2) & (3)
⇒ ln y = ln 5x + ln C
⇒ ln y = ln C5x
⇒ y = C × 5x ------(1)
According to the question, y(0) = ln 5
ln 5 = C × 50
⇒ C = ln 5
Put this value in equation (1)
⇒ y = 5x ln 5
Put x = 1
∴ y(1) = 5 ln 5
Last updated on May 30, 2025
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