Solve the initial value problem given by the differential equation
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{4} y^{2}
$$
together with the initial condition: if $x=-2$ then $y=2$.
To solve this, rearrange and integrate to get
$$
\int f(y) \mathrm{d} y=\int g(x) \mathrm{d} x
$$
where
$$
f(y), g(x)=
$$
$\square$
Enter your responses separated by a comma.

The solution to the original initial value problem is
$$
y=
$$
$\square$

Question

Solve the initial value problem given by the differential equation
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{4} y^{2}
$$
together with the initial condition: if $x=-2$ then $y=2$.
To solve this, rearrange and integrate to get
$$
\int f(y) \mathrm{d} y=\int g(x) \mathrm{d} x
$$
where
$$
f(y), g(x)=
$$
$\square$
Enter your responses separated by a comma.

The solution to the original initial value problem is
$$
y=
$$
$\square$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Separate variables by dividing both sides by $ y^2 $ and multiplying both sides by $ dx $. ‖ ‖ step 2 ⋮ We get $ \frac{1}{y^2} dy = x^4 dx $. ‖ ‖ step 3 ⋮ Integrate both sides: $ \int \frac{1}{y^2} dy = \int x^4 dx $. ‖ ‖ step 4 ⋮ The integral of $ \frac{1}{y^2} $ with respect to $ y $ is $ -\frac{1}{y} $, and the integral of $ x^4 $ with respect to $ x $ is $ \frac{x^5}{5} $. ‖ ‖ step 5 ⋮ After integration, we have $ -\frac{1}{y} = \frac{x^5}{5} + C $. ‖ ‖ step 6 ⋮ Use the initial condition $ x = -2 $ and $ y = 2 $ to solve for $ C $. ‖ ‖ step 7 ⋮ Substitute the initial condition into the integrated equation: $ -\frac{1}{2} = \frac{(-2)^5}{5} + C $. ‖ ‖ step 8 ⋮ Solve for $ C $: $ C = -\frac{1}{2} - \frac{-32}{5} = -\frac{1}{2} + \frac{32}{5} = \frac{25}{10} - \frac{5}{10} = \frac{20}{10} = 2 $. ‖ ‖ step 9 ⋮ Substitute $ C $ back into the integrated equation to get the solution: $ -\frac{1}{y} = \frac{x^5}{5} + 2 $. ‖ ‖ step 10 ⋮ Solve for $ y $: $ y = -\frac{1}{\left(\frac{x^5}{5} + 2\right)} $. ‖ ∻Answer∻ ⚹ $ f(y) = \frac{1}{y^2}, g(x) = x^4 $ ⚹ ⚹ $ y = -\frac{1}{\left(\frac{x^5}{5} + 2\right)} $ ⚹ ∻Key Concept∻ ⚹Separation of Variables and Integration⚹ ∻Explanation∻ ⚹To solve the differential equation, we separated variables and integrated both sides. We then used the initial condition to find the constant of integration and solve for $ y $.⚹◊What are the steps to solve the initial value problem given by the differential equation $\frac{dy}{dx} = x^4 y^2$ with the initial condition $x=-2$ and $y=2$?⍭ Generate me a similar question◊

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