Answer :
To find the general solution to the differential equation [tex]\(\frac{d y}{d x} = \csc^2 x - 12 x^5\)[/tex], we need to integrate each term separately. Let's go through the steps:
1. Identify the Differential Equation:
We have [tex]\(\frac{d y}{d x} = \csc^2 x - 12 x^5\)[/tex].
2. Separate the Terms:
We can rewrite this as:
[tex]\[
\frac{d y}{d x} = \csc^2 x - 12 x^5
\][/tex]
Our task is to integrate each part with respect to [tex]\(x\)[/tex].
3. Integrate [tex]\(\csc^2 x\)[/tex]:
The integral of [tex]\(\csc^2 x\)[/tex] with respect to [tex]\(x\)[/tex] is known to be [tex]\(-\cot x\)[/tex]. This is a standard trigonometric integral.
4. Integrate [tex]\(-12 x^5\)[/tex]:
The integral of [tex]\(-12 x^5\)[/tex] with respect to [tex]\(x\)[/tex] is calculated by finding the antiderivative:
[tex]\[
\int -12 x^5 \, dx = -2 x^6
\][/tex]
This is done by increasing the exponent by one (to 6) and dividing by the new exponent.
5. Combine the Results:
After integrating both terms, we combine them to get the general solution of the differential equation:
[tex]\[
y = -\cot x - 2 x^6 + C
\][/tex]
where [tex]\(C\)[/tex] is an arbitrary constant representing the constant of integration.
Thus, the general solution to the differential equation is:
[tex]\[
y = -\cot x - 2 x^6 + C
\][/tex]
1. Identify the Differential Equation:
We have [tex]\(\frac{d y}{d x} = \csc^2 x - 12 x^5\)[/tex].
2. Separate the Terms:
We can rewrite this as:
[tex]\[
\frac{d y}{d x} = \csc^2 x - 12 x^5
\][/tex]
Our task is to integrate each part with respect to [tex]\(x\)[/tex].
3. Integrate [tex]\(\csc^2 x\)[/tex]:
The integral of [tex]\(\csc^2 x\)[/tex] with respect to [tex]\(x\)[/tex] is known to be [tex]\(-\cot x\)[/tex]. This is a standard trigonometric integral.
4. Integrate [tex]\(-12 x^5\)[/tex]:
The integral of [tex]\(-12 x^5\)[/tex] with respect to [tex]\(x\)[/tex] is calculated by finding the antiderivative:
[tex]\[
\int -12 x^5 \, dx = -2 x^6
\][/tex]
This is done by increasing the exponent by one (to 6) and dividing by the new exponent.
5. Combine the Results:
After integrating both terms, we combine them to get the general solution of the differential equation:
[tex]\[
y = -\cot x - 2 x^6 + C
\][/tex]
where [tex]\(C\)[/tex] is an arbitrary constant representing the constant of integration.
Thus, the general solution to the differential equation is:
[tex]\[
y = -\cot x - 2 x^6 + C
\][/tex]
