Question Statement

Eliminate the arbitrary constants from the following equations:

(i) $y=a{e}^x+b{e}^{-x}+c$

(ii) $y=\cos (x+b)$

(iii) $y=mx+c$

(iv) $y=b{x}^2+2ax$

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Background and Explanation

To eliminate arbitrary constants, we differentiate the given equation with respect to the independent variable ($x$). The number of times we differentiate usually corresponds to the number of arbitrary constants present. We then use substitution or further differentiation to form a differential equation that no longer contains those constants.

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Solution

Part (i)

Given the equation: $y=a{e}^x+b{e}^{-x}+c$

Differentiate with respect to $x$: $y^{\prime }=a{e}^x+b{e}^{-x}\cdot (-1)+0$ $y^{\prime }=a{e}^x-b{e}^{-x}$

Differentiate again with respect to $x$: $y^{\prime \prime }=a{e}^x-b{e}^{-x}\cdot (-1)$ $y^{\prime \prime }=a{e}^x+b{e}^{-x}$

From equation (1), we know that $y-c=a{e}^x+b{e}^{-x}$. However, to fully eliminate the constants without needing to solve for $c$, we differentiate equation (3) once more: $y^{\prime \prime \prime }=a{e}^x-b{e}^{-x}$

By comparing this result to equation (2), we can see that $y^{\prime \prime \prime }=y^{\prime }$. $y^{\prime \prime \prime }-y^{\prime }=0$

Part (ii)

Given the equation: $y=\cos (x+b)$

Differentiate with respect to $x$: $y^{\prime }=-\sin (x+b)\cdot 1$

Differentiate again with respect to $x$: $y^{\prime \prime }=-\cos (x+b)$

Using equation (1), we substitute $y$ back into the equation: $y^{\prime \prime }=-y$ $y^{\prime \prime }+y=0$

Part (iii)

Given the equation: $y=mx+c$

Differentiate with respect to $x$: $y^{\prime }=m+0$ $y^{\prime }=m$

Differentiate again with respect to $x$ to eliminate the remaining constant $m$: $y^{\prime \prime }=0$

Part (iv)

Given the equation: $y=b{x}^2+2ax$

Differentiate with respect to $x$: $y^{\prime }=b(2x)+2a\cdot 1$ $y^{\prime }=2bx+2a$

Differentiate again with respect to $x$: $y^{\prime \prime }=2b\cdot (1)+0$ $y^{\prime \prime }=2b$

Differentiate a third time to eliminate the final constant $b$: $y^{\prime \prime \prime }=0$

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Key Formulas or Methods Used

• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Exponential Derivative: $\frac{d}{dx}(e^{ax})=a{e}^{ax}$
• Trigonometric Derivatives: $\frac{d}{dx}(\cos x)=-\sin x$ and $\frac{d}{dx}(\sin x)=\cos x$
• Elimination Method: Differentiating an equation $n$ times to eliminate $n$ arbitrary constants.

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Summary of Steps

1. Identify Constants: Count the number of arbitrary constants in the equation.
2. Differentiate: Perform successive differentiation with respect to $x$.
3. Substitute: Look for opportunities to substitute the original function $y$ or its lower-order derivatives back into the higher-order derivative equations.
4. Final Form: Arrange the resulting terms into a differential equation where no arbitrary constants ($a,b,c,m$, etc.) remain.