Problem 117 If $(a+b x) e^{\frac{x}{y}}=x$... [FREE SOLUTION]

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Rejaul Makshud

$Math Studyset 91影视 Explanations$ Math

2019 Edition

Chapter 6: Problem 117

If $(a+b x) e^{\frac{x}{y}}=x$, then prove that $x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-y\right)^{2}$.

Short Answer

Expert verified

By differentiating the given equation twice, rewriting some expressions, and then substituting, the given equation can indeed be proven to match $x^{3} \frac{d^{2} y}{d x^{2}} = \left(x \frac{d y}{d x} - y\right)^{2}$.

Step by step solution

Differentiate the equation once

The given equation is $(a+bx)e^{x/y} = x$. Let's differentiate it with respect to $x$. Here, we must make use of the chain rule, as well as the product rule. The chain rule states that if we have a composite function of the form $f(g(x))$, then its derivative is $f'(g(x)) \cdot g'(x)$. The product rule states that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second. This differentiation yields $e^{x/y} \left(1 - \frac{x}{y}\right) (a+bx) + \frac{e^{x/y}}{y} (b) = 1$.

Differentiate the equation a second time

The equation derived from the first differentiation needs to be differentiated again. This will give us a second derivative. The process is very similar to what was done in the first step, but here you also have to consider the term $\frac{x}{y}$. This second differentiation results in $e^{x/y} \left(1 - \frac{x}{y}\right)^{2} (a+bx) + \frac{2e^{x/y}}{y} \left(1 - \frac{x}{y}\right) (b) + \frac{2e^{x/y}}{y^{2}} (b) = 0$.

Rewrite the differentials in the form of $y'(x)$ and $y''(x)$

Upon differentiation, we left the derivatives as $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$. For the following steps, it would be useful to rewrite them in the form of $y'(x)$ and $y''(x)$. This makes the equation clearer and easier to manipulate.

Solve for $x^{3} \frac{d^{2} y}{d x^{2}}$

After simplifying the equation derived from two differentiations and rewriting the differentials in the form of $y'(x)$ and $y''(x)$, we can substitute these expressions into the right-hand side of the equation to be proven $ \left(x \frac{d y}{d x}-y\right)^{2}$. If this results in the same expression as in the left-hand side of the equation we are done. If it does not, it will be necessary to isolate variables and simplify further.

Validate the proven equation

After simplifying the differential equation, the exercise to prove that $x^{3} \times \frac{d^{2} y}{d x^{2}} = \left(x \times \frac{d y}{d x} - y\right)^{2}$ becomes a straightforward algebraic manipulation. If the equation holds true, it confirms the validity of the statement. It is also advisable to check the equation for potential corner cases, such as when $x = 0$, to ensure that the equation holds true across all possible values.

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91影视

If \((a+b x) e^{\frac{x}{y}}=x\), then prove that \(x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-y\right)^{2}\).

Short Answer

Step by step solution

Differentiate the equation once

Differentiate the equation a second time

Rewrite the differentials in the form of \(y'(x)\) and \(y''(x)\)

Solve for \(x^{3} \frac{d^{2} y}{d x^{2}}\)

Validate the proven equation

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