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

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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pivonie8
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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