To solve the given problem, we analyze the equation provided:

The equation is:

$y = C_1x + C_2x^2 + C_3x^3$

We need to eliminate the arbitrary constants $C_1, C_2,$ and $C_3$ by differentiating and manipulating the equation.

Step 1: Differentiate the equation with respect to $x$.

$\frac{dy}{dx} = C_1 + 2C_2x + 3C_3x^2$

Step 2: Differentiate again to find the second derivative.

$\frac{d^2y}{dx^2} = 2C_2 + 6C_3x$

Step 3: Differentiate once more to find the third derivative.

$\frac{d^3y}{dx^3} = 6C_3$

Step 4: Use the derivatives to eliminate $C_1, C_2,$ and $C_3$.

From the third derivative: $\frac{d^3y}{dx^3} = 6C_3$

Since $\frac{d^3y}{dx^3}$ is constant, the third derivative contains no further variables or constants. By back-substitution, $C_3$, $C_2$, and $C_1$ are effectively eliminated in the final differential equation.

Final Result:

The differential equation is: $\frac{d^3y}{dx^3} = 0$

This equation represents the elimination of the arbitrary constants $C_1, C_2,$ and $C_3$.

Do you need further clarification or additional steps explained? Let me know!

5 Related Questions:

1. What is the process of determining the order of a differential equation from a given expression?
2. How do arbitrary constants relate to the general solution of a differential equation?
3. Can a higher-order differential equation always be reduced to an equation with no arbitrary constants?
4. What are some examples of physical problems where such polynomial equations arise?
5. How would the solution change if more arbitrary constants were included in the original equation?

Tip:

When eliminating constants from an equation, always differentiate just enough times to isolate each constant before substituting back into the equation!