Certainly! Let's solve the differential equation step-by-step:
Given the differential equation:
[tex]\[
\frac{d y}{d x} = \frac{6 x^5 - 2 x^3 + 6}{6 y}
\][/tex]
To solve this differential equation, we can use the method of separation of variables. This involves rearranging the equation so that one side contains only [tex]\( y \)[/tex] and its differential [tex]\( dy \)[/tex], and the other side contains only [tex]\( x \)[/tex] and its differential [tex]\( dx \)[/tex].
1. First, we multiply both sides by [tex]\( 6y \)[/tex] to get rid of the denominator on the right side.
[tex]\[
6 y \frac{d y}{d x} = 6 x^5 - 2 x^3 + 6
\][/tex]
2. Now, we can separate the variables [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
6 y \, dy = (6 x^5 - 2 x^3 + 6) \, dx
\][/tex]
3. Integrate both sides:
[tex]\[
\int 6 y \, dy = \int (6 x^5 - 2 x^3 + 6) \, dx
\][/tex]
4. Calculate the integrals:
[tex]\[
6 \int y \, dy = \int 6 x^5 \, dx - \int 2 x^3 \, dx + \int 6 \, dx
\][/tex]
5. Perform the integration:
[tex]\[
6 \left( \frac{y^2}{2} \right) = 6 \left( \frac{x^6}{6} \right) - 2 \left( \frac{x^4}{4} \right) + 6x + C
\][/tex]
Simplifying the equations:
[tex]\[
3 y^2 = x^6 - \frac{x^4}{2} + 6x + C
\][/tex]
6. To isolate [tex]\( y \)[/tex], we solve for [tex]\( y \)[/tex]:
[tex]\[
y^2 = \frac{1}{3} \left( x^6 - \frac{x^4}{2} + 6x + C \right)
\][/tex]
[tex]\[
y = \pm \sqrt{\frac{1}{3} \left( x^6 - \frac{x^4}{2} + 6x + C \right)}
\][/tex]
Thus, the general solution to the differential equation [tex]\(\frac{d y}{d x} = \frac{6 x^5 - 2 x^3 + 6}{6 y}\)[/tex] is:
[tex]\[
y = \pm \sqrt{\frac{1}{3} \left( x^6 - \frac{x^4}{2} + 6x + C \right)}
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
