Finding the Equation of a Curve from its Derivative

The problem asks us to find the equation of a curve given its derivative, \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\), and a specific point, (-1, -2), that the curve passes through. This is a classic problem involving solving a differential equation.

The given differential equation is:

\[ \frac{{{\rm{dy}}}}{{{\rm{dx}}}} = {\rm{}} - {{\rm{x}}^2} - \frac{1}{{{{\rm{x}}^3}}} \]

To find the equation of the curve \(y(x)\), we need to integrate the given derivative with respect to \(x\).

First, we separate the variables:

\[ {\rm{dy}} = \left( { - {{\rm{x}}^2} - \frac{1}{{{{\rm{x}}^3}}}} \right){\rm{dx}} \]

Now, we integrate both sides:

\[ \int {{\rm{dy}}} = \int {\left( { - {{\rm{x}}^2} - \frac{1}{{{{\rm{x}}^3}}}} \right){\rm{dx}}} \]

We can integrate term by term:

\[ y = \int { - {{\rm{x}}^2}{\rm{dx}}} - \int {{{\rm{x}}^{ - 3}}{\rm{dx}}} \]

Using the power rule for integration, \(\int {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C\) (for \(n \neq -1\)):

• The integral of \(-{{\rm{x}}^2}\) is \( - \frac{{{x^{2 + 1}}}}{{2 + 1}} = - \frac{{{x^3}}}{3} \).
• The integral of \(-{{\rm{x}}^{ - 3}}\) is \( - \frac{{{x^{ - 3 + 1}}}}{{ - 3 + 1}} = - \frac{{{x^{ - 2}}}}{{ - 2}} = \frac{{{x^{ - 2}}}}{2} = \frac{1}{{2{{\rm{x}}^2}}} \).

Combining the integrals and adding the constant of integration, \(C\):

\[ y = - \frac{{{x^3}}}{3} + \frac{1}{{2{{\rm{x}}^2}}} + C \]

This equation represents a family of curves that satisfy the differential equation. To find the specific curve that passes through the point (-1, -2), we substitute \(x = -1\) and \(y = -2\) into the equation:

\[ - 2 = - \frac{{{{\left( { - 1} \right)}^3}}}{3} + \frac{1}{{2{{\left( { - 1} \right)}^2}}} + C \]

Let's simplify this expression:

• \({\left( { - 1} \right)^3} = - 1\)
• \({\left( { - 1} \right)^2} = 1\)

Substituting these values:

\[ - 2 = - \frac{{\left( { - 1} \right)}}{3} + \frac{1}{{2\left( 1 \right)}} + C \]

\[ - 2 = \frac{1}{3} + \frac{1}{2} + C \]

To find \(C\), we need to isolate it. First, combine the fractions on the right side by finding a common denominator, which is 6:

\[ - 2 = \frac{2}{6} + \frac{3}{6} + C \]

\[ - 2 = \frac{5}{6} + C \]

Now, solve for \(C\):

\[ C = - 2 - \frac{5}{6} \]

\[ C = - \frac{{12}}{6} - \frac{5}{6} \]

\[ C = - \frac{{17}}{6} \]

Now substitute the value of \(C\) back into the equation of the curve:

\[ y = - \frac{{{x^3}}}{3} + \frac{1}{{2{{\rm{x}}^2}}} - \frac{{17}}{6} \]

To match the format of the given options, we can clear the denominators by multiplying the entire equation by the least common multiple of 3, \(2x^2\), and 6, which is \(6x^2\):

\[ 6{{\rm{x}}^2}y = 6{{\rm{x}}^2}\left( { - \frac{{{x^3}}}{3}} \right) + 6{{\rm{x}}^2}\left( {\frac{1}{{2{{\rm{x}}^2}}}} \right) - 6{{\rm{x}}^2}\left( {\frac{{17}}{6}} \right) \]

Simplify each term:

• \(6{{\rm{x}}^2}\left( { - \frac{{{x^3}}}{3}} \right) = - 2{{\rm{x}}^{2 + 3}} = - 2{{\rm{x}}^5}\)
• \(6{{\rm{x}}^2}\left( {\frac{1}{{2{{\rm{x}}^2}}}} \right) = \frac{{6{{\rm{x}}^2}}}{{2{{\rm{x}}^2}}} = 3\)
• \(6{{\rm{x}}^2}\left( { - \frac{{17}}{6}} \right) = - 17{{\rm{x}}^2}\)

So the equation becomes:

\[ 6{{\rm{x}}^2}y = - 2{{\rm{x}}^5} + 3 - 17{{\rm{x}}^2} \]

Rearrange the terms to set the equation equal to zero and match the option format:

\[ 6{{\rm{x}}^2}y + 2{{\rm{x}}^5} + 17{{\rm{x}}^2} - 3 = 0 \]

This can also be written as:

\[ 6{{\rm{x}}^2}y + 17{{\rm{x}}^2} + 2{{\rm{x}}^5} - 3 = 0 \]

Comparing this result with the given options, we find it matches one of them.

The equation of the curve passing through (-1, -2) is \(6{{\rm{x}}^2}y + 17{{\rm{x}}^2} + 2{{\rm{x}}^5} - 3 = 0\).

Revision Table: Solving Differential Equations

Additional Information on Finding Curve Equations

Finding the equation of a curve from its slope function (the derivative) is a fundamental application of integration in calculus. The derivative \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) gives the slope of the tangent line to the curve at any point \((x, y)\).

When you are given \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = f(x)\), integrating \(f(x)\) with respect to \(x\) gives you \(y = \int f(x) dx = F(x) + C\). The constant \(C\) means there are infinitely many curves with the same derivative; they are vertical shifts of each other. For example, \(y = x^2\), \(y = x^2 + 5\), and \(y = x^2 - 3\) all have the same derivative, \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = 2x\).

A specific point that the curve must pass through is called an initial condition or a boundary condition. This condition provides the necessary information to determine the exact value of \(C\), thus identifying the unique curve from the family of solutions.

In this problem, \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) was given as a function of \(x\) only, making it a separable differential equation that is straightforward to solve by direct integration. For more complex differential equations, other techniques like substitution, integrating factors, or series solutions might be required.