Question

The differential equation representing the family of curves \(\mathrm{y}^{2}=2 \mathrm{c}(\mathrm{x}+\mathrm{v} \mathrm{c})\), where \(\mathrm{c}\) is a positive parameter, is of order and degree as follows. (A) order 2, degree 1 (B) order 1 , degree 2 (C) order 2 , degree 2 (D) order 1 , degree 3

Short answer

The given equation is \(y^2 = 2c(x+vc)\), where \(c\) and \(v\) are arbitrary constants. In order to find the differential equation representing this family of curves, first eliminate these constants. After differentiating the equation with respect to \(x\), we obtain the differential equation \(0=(yy'(x+vc) - y^2)/(x+vc)^2\). The order of the differential equation is 1 (represented by \(y'\)), and the degree is also 1. So, none of the given options A, B, C, D are correct.

Step-by-step solution

Express the equation as a differential equation

The given equation is \(y^2=2c(x+vc)\). Here \(v\) and \(c\) are arbitrary constants. We need to eliminate these constants from the equation to express it as a differential equation. First replace \(c\) using \(y^2/2(x+vc)\) and differentiate the resulting equation with respect to \(x\) to eliminate \(v\).

Differentiate the equation

Start with the equation \(c = \frac{y^2}{2(x+vc)}\). Differentiate this equation with respect to \(x\) using chain rule: \(\frac{d}{dx}c = \frac{d}{dx}[y^2/2(x+vc)]\) Since \(c\) is a constant, it's differentiation will be zero and on the other side of the equation using quotient and chain rule: \(0=\frac{[(2yy')(2(x+vc)) - y^2(2+2vc')]}{4(x+vc)^2}\) Considering that the derivative of \(c\) with respect to \(x\) is zero (\(c' = 0\)), the equation simplifies to: \(0=\frac{(2yy')(2x+2vc) - 2y^2}{4(x+vc)^2}\) Then after simplifying it we get: \(0=(yy'[x+vc] - y^2)/(x+vc)^2\)

Identify the order and degree

The revised equation is a differential equation. We can see the order and degree of this differential equation. Order of a differential equation is basically the order of the highest order derivative present in the equation. Degree is the power of the highest order derivative in the equation only if the differential coefficients in the equation are free from radicals and fractions. Here in \(0=(yy'[x+vc] - y^2)/(x+vc)^2\), the highest order of derivative is 1 (which is \(y'\)). The power of this highest order derivative is also 1 since it is not under any radical or fractional power. Hence, the order of the differential equation is 1 and the degree is 1. So, none of the given options A, B, C, D are correct.

Key concepts

Order of Differential Equation

In differential equations, the concept of order is crucial because it tells us the highest derivative present in the equation. Understanding the order helps to determine the complexity involved in solving the differential equation.

Here are some key points:

• Order is determined by the highest derivative, not by the number of derivatives present.
• The simplest form of differential equations are first order, which involve only the first derivative.
• Higher order equations, like second order or third order, involve second or third derivatives, respectively.

This information is imperative because higher order equations typically require more complicated methods to solve, involving additional integration steps or specialized formulas.

Degree of Differential Equation

The degree of a differential equation refers to the power to which the highest order derivative is raised in the equation, provided the equation is polynomial in derivatives. This means that for a degree to be well-defined, the equation should not contain derivatives in any form other than polynomial.

Let's break it down:

• The degree is only defined when derivatives appear in polynomial form—no fractions or roots.
• If the differential equation has expressions like square roots of derivatives or derivatives in denominators, the degree is not defined.
• A first order equation may have a higher degree depending on the highest power of the first derivative.

In our example, the equation simplifies to a polynomial form, making the degree straightforward to identify.

Chain Rule

The chain rule is a fundamental tool in calculus used for differentiating composite functions, allowing one to differentiate an outer function with respect to an inner function. It is particularly useful in deriving expressions in differential equations where functions are nested.

Here's how the chain rule works:

• If you have a function within a function, such as \(f(g(x))\), the derivative is calculated as \(f'(g(x)) \cdot g'(x)\).
• This rule simplifies the differentiation process for complex, nested functions, often seen in physics and engineering problems.
• When applying the chain rule, always differentiate the outer function first, then multiply by the derivative of the inner function.

In our example with the family of curves, using the chain rule helped to efficiently differentiate and simplify the equation, ultimately aiding in the determination of the order and degree of the resulting differential equation.