Final answer:
The general solution of the differential equation y' = 7x⁶y + x⁶ involves finding an integrating factor, which is e^{-x^7}, and then integrating the transformed equation to obtain the solution y(x) in terms of an indefinite integral and a constant of integration C.
Step-by-step explanation:
To find the general solution of the differential equation y' = 7x⁶y + x⁶, we need to solve it using an appropriate method. Since the equation is first order, we might look for an integrating factor or notice that it is a first-order linear differential equation. The equation can be written in the standard linear form:
y' - 7x⁶y = x⁶.
The integrating factor, μ(x), is given by:
μ(x) = e^{∫( -7x⁶) dx} = e^{-7x^7/7} = e^{-x^7}.
Multiplying through by the integrating factor, we get:
μ(x)y' - μ(x)7x⁶y = μ(x)x⁶.
Now the left side of the equation is the derivative of μ(x)y with respect to x:
(μ(x)y)' = e^{-x^7}x⁶.
Integrating both sides with respect to x:
∫ (μ(x)y)' dx = ∫ e^{-x^7}x⁶ dx.
this results in:
μ(x)y = ∫ e^{-x^7}x⁶ dx + C, where C is the constant of integration.
To find the particular solution, you would have to integrate e^{-x^7}x⁶ and then multiply by e^{x^7}. However, without specific initial conditions, we can only express the general solution in terms of this indefinite integral.
Therefore, the general solution is:
y(x) = e^{x^7} • (∫ e^{-x^7}x⁶ dx + C).