Key Concepts and Formulas

• Formation of Differential Equations: The goal is to eliminate arbitrary constants from a given equation by differentiation and algebraic manipulation. The order of the resulting differential equation equals the number of arbitrary constants.
• Chain Rule: $\frac{d}{dx}e^{f(x)} = e^{f(x)} \cdot f'(x)$.
• Elimination of Constants: With $n$ arbitrary constants, differentiate $n$ times to obtain $n+1$ equations. Then, eliminate the constants using algebraic manipulation.

Step-by-Step Solution

Step 1: Identify the given equation and arbitrary constants.

We are given the family of curves: $y = c_1 e^{c_2 x}$ Here, $c_1$ and $c_2$ are the arbitrary constants. Since there are two arbitrary constants, we need to differentiate twice and then eliminate $c_1$ and $c_2$.

Step 2: Differentiate with respect to $x$ (First Derivative).

We differentiate the given equation with respect to $x$ to find $y'$: $y' = \frac{d}{dx}(c_1 e^{c_2 x})$ Using the chain rule, we get: $y' = c_1 c_2 e^{c_2 x} \quad (1)$ This equation now contains $y'$, $c_1$, and $c_2$.

Step 3: Differentiate with respect to $x$ (Second Derivative).

We differentiate equation (1) with respect to $x$ to find $y''$: $y'' = \frac{d}{dx}(c_1 c_2 e^{c_2 x})$ Using the chain rule again: $y'' = c_1 c_2^2 e^{c_2 x} \quad (2)$ We now have three equations: the original equation and equations (1) and (2).

Step 4: Eliminate $c_1$ and $c_2$.

From equation (1), we can express $c_1 e^{c_2 x}$ as: $c_1 e^{c_2 x} = \frac{y'}{c_2}$ Substituting this into the original equation $y = c_1 e^{c_2 x}$, we get: $y = \frac{y'}{c_2}$ Thus, $c_2 = \frac{y'}{y} \quad (3)$

Now, substitute equation (3) into equation (2): $y'' = c_1 \left( \frac{y'}{y} \right)^2 e^{c_2 x}$ $y'' = c_1 e^{c_2 x} \left( \frac{y'}{y} \right)^2$ Since $y = c_1 e^{c_2 x}$, we substitute $y$ for $c_1 e^{c_2 x}$: $y'' = y \left( \frac{y'}{y} \right)^2$ $y'' = y \frac{(y')^2}{y^2}$ $y'' = \frac{(y')^2}{y}$ Multiplying both sides by $y$, we get: $yy'' = (y')^2$

Step 5: Check the Correct Answer We are told that the answer is y'' = y'y. But we arrived at yy'' = (y')^2. Let's re-examine our steps, focusing on the final elimination. We seek to arrive at y'' = y'y, or equivalently, y''/y' = y'.

From Step 2, we have $y' = c_1 c_2 e^{c_2 x}$. From Step 3, we have $y'' = c_1 c_2^2 e^{c_2 x}$. Dividing the second equation by the first equation, we obtain: $\frac{y''}{y'} = \frac{c_1 c_2^2 e^{c_2 x}}{c_1 c_2 e^{c_2 x}} = c_2$ So, $c_2 = \frac{y''}{y'}$.

Now, from Step 1, we have $y = c_1 e^{c_2 x}$. Taking the natural logarithm of both sides gives: $\ln y = \ln c_1 + c_2 x$ Differentiating with respect to $x$ gives: $\frac{y'}{y} = c_2$ Therefore, $\frac{y'}{y} = \frac{y''}{y'}$, which gives $(y')^2 = yy''$. However, it is stated that the correct answer is $y'' = y'y$. Let's re-examine: From $y = c_1 e^{c_2 x}$, we have $y' = c_1 c_2 e^{c_2 x}$ and $y'' = c_1 c_2^2 e^{c_2 x}$. Then, $\frac{y'}{y} = c_2$ and $\frac{y''}{y'} = c_2$. So, $\frac{y'}{y} = \frac{y''}{y'}$. This implies $(y')^2 = yy''$. Now, since the stated correct answer is $y'' = y'y$, we examine the ratio: $\frac{y''}{y'} = y$. Since $\frac{y''}{y'} = c_2$, we have $c_2 = y$. But $\frac{y'}{y} = c_2$, so $\frac{y'}{y} = y$, or $y' = y^2$.

Let's try another approach. $y = c_1 e^{c_2 x}$. $y' = c_1 c_2 e^{c_2 x}$. $y'' = c_1 c_2^2 e^{c_2 x}$. We have $y' = c_2 y$ and $y'' = c_2 y' = c_2 (c_2 y) = c_2^2 y$. Then $c_2 = \frac{y'}{y}$, so $y'' = (\frac{y'}{y}) y' = \frac{(y')^2}{y}$. Thus $yy'' = (y')^2$. We are told the correct answer is $y'' = y'y$. Then $y''/y' = y$. Since $y' = c_2 y$, we have $c_2 = y'/y$. Also, $y'' = c_2 y' = (y'/y) y' = (y')^2 / y$. This does not lead to option A.

We have $y'' = c_2 y'$, so $c_2 = y''/y'$. Also, $y' = c_2 y$, so $c_2 = y'/y$. Then $y''/y' = y'/y$, so $y y'' = (y')^2$. Let's look at the ratios. $y'/y = c_2$ and $y''/y' = c_2$. So $y'/y = y''/y'$. Then $y (y'') = (y')^2$.

The correct option is (C). There is an error in the stated answer.

Common Mistakes & Tips

• Algebraic Errors: Be extremely careful with algebraic manipulations when eliminating constants. A small error can lead to an incorrect differential equation.
• Order of Differentiation: Ensure you differentiate the correct number of times based on the number of arbitrary constants.
• Chain Rule: Remember to apply the chain rule correctly when differentiating exponential functions.

Summary

The problem requires us to find the differential equation representing a family of curves with two arbitrary constants. We differentiate the given equation twice, obtaining three equations in total. Then, we eliminate the arbitrary constants $c_1$ and $c_2$ through algebraic manipulation. After careful steps, we arrive at the differential equation $yy'' = (y')^2$. The solution indicates an error in the problem statement or answer key. The correct answer should be option (C).

Final Answer

The final answer is \boxed{yy'' = (y')^2}, which corresponds to option (C).