Find the general solution of the differential equation or state that the differe

Chardonnay Felix 2021-09-21 Answered
Find the general solution of the differential equation or state that the differential equation is not separable.
\(\displaystyle{y}'={x}^{{{6}}}{y}\)

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Expert Answer

pivonie8
Answered 2021-09-22 Author has 5672 answers
Step 1
Given differential equation:
\(\displaystyle{y}'={x}^{{{6}}}{y}\)
Convert the above differential equation in the first order separable ordinary differential equation
\(\displaystyle{\frac{{{y}'}}{{{y}}}}={x}^{{{6}}}\)
Step 2
Integrate the both sides, we get
\(\displaystyle\int{\frac{{{\left.{d}{y}\right.}}}{{{y}}}}=\int{x}^{{{6}}}{\left.{d}{x}\right.}\)
We know that \(\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}}+{c}\)
where, "c" is integrating constant.
\(\displaystyle{\ln{{\left({y}\right)}}}={\frac{{{x}^{{{7}}}}}{{{7}}}}+{c}\)
\(\displaystyle{y}={e}^{{{\frac{{{x}^{{{7}}}}}{{{7}}}}+{c}}}\)
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