Integrate each side of the equation by x. Antiderivatives will differ by an arbitrary function C1(y):
Repeat this:
C2(y) is an arbitrary function.
Integrate each side of the equation by x. Antiderivatives will differ by an arbitrary function C1(y):
This is linear ordinary differential linear equation by y. Then by the well-known formula
[https://en.wikipedia.org/wiki/Linear_differential_equation#First-order_equation_with_variable_coefficients]
where y0 is any preselected number (let y0=0 for example), C2(x) is an arbitrary function of x.
Then
Now let the integrals not depend on x: this could be taken into account in arbitrary C2(x). Then consider a new arbitrary function C3(y) based on arbitrary function C1(y):
"C_3(y) = \ne^y\\int\\limits_{y_0}^{y}{C_1(t)e^{-t}dt},"and a new arbitrary function C4(x) based on arbitrary function C2(x):
"\\int\\limits_{0}^{y}{e^{-t}dt} = -e^{-y} + 1,""C_4(x) = 6(x-1)e^x + C_2(x)."Then the expression is simplified:
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