Degree of the differential equation \( \frac{d^2y}{dx^2} + 3 \left( \frac{dy}{dx} \right)^{\frac{1}{2}} = y^2 + e^x \) is:
2
Understanding Order and Degree of Differential Equations
To determine the degree of a differential equation, we first need to understand what order and degree mean.
- The order of a differential equation is the order of the highest derivative appearing in the equation.
- The degree of a differential equation is the highest power of the highest order derivative after the equation has been made free from radicals and fractions involving the derivatives.
Let's find the degree of the given differential equation: \( \frac{d^2y}{dx^2} + 3 \left( \frac{dy}{dx} \right)^{\frac{1}{2}} = y^2 + e^x \)
Analyzing the Given Differential Equation
The given differential equation is:
\( \frac{d^2y}{dx^2} + 3 \left( \frac{dy}{dx} \right)^{\frac{1}{2}} = y^2 + e^x \)
Let's identify the highest order derivative. The derivatives present are \( \frac{dy}{dx} \) (first order) and \( \frac{d^2y}{dx^2} \) (second order). The highest order derivative is \( \frac{d^2y}{dx^2} \), so the order of this differential equation is 2.
Clearing Radicals for Degree Calculation
Now, we need to find the degree. The equation contains a term with a radical involving a derivative: \( \left( \frac{dy}{dx} \right)^{\frac{1}{2}} \), which is the square root of the first derivative. To find the degree, the equation must be a polynomial in its derivatives. We need to eliminate this radical.
Let's isolate the term with the radical:
\( 3 \left( \frac{dy}{dx} \right)^{\frac{1}{2}} = y^2 + e^x - \frac{d^2y}{dx^2} \)
To remove the square root (power of \( \frac{1}{2} \)), we square both sides of the equation:
\( \left[ 3 \left( \frac{dy}{dx} \right)^{\frac{1}{2}} \right]^2 = \left[ y^2 + e^x - \frac{d^2y}{dx^2} \right]^2 \)
This simplifies to:
\( 9 \left( \frac{dy}{dx} \right) = \left( y^2 + e^x - \frac{d^2y}{dx^2} \right)^2 \)
Finding the Degree from the Polynomial Form
The differential equation is now in a form where the derivatives appear as powers, without radicals or fractions involving them. We need to find the highest power of the highest order derivative in this equation.
The highest order derivative is \( \frac{d^2y}{dx^2} \). In the equation \( 9 \left( \frac{dy}{dx} \right) = \left( y^2 + e^x - \frac{d^2y}{dx^2} \right)^2 \), the highest order derivative \( \frac{d^2y}{dx^2} \) is inside the term \( \left( y^2 + e^x - \frac{d^2y}{dx^2} \right)^2 \). When this term is expanded, the highest power of \( \frac{d^2y}{dx^2} \) will be \( (\frac{d^2y}{dx^2})^2 \).
Thus, the highest power of the highest order derivative \( \frac{d^2y}{dx^2} \) is 2.
Therefore, the degree of the differential equation is 2.
Summary of Steps
- Identify the highest order of derivative (which is 2 for \( \frac{d^2y}{dx^2} \)). The order is 2.
- Check for radicals or fractions involving derivatives. The term \( \left( \frac{dy}{dx} \right)^{\frac{1}{2}} \) is present.
- Clear the radical by algebraic manipulation (isolating and squaring both sides).
- Identify the highest order derivative in the resulting polynomial form. It is still \( \frac{d^2y}{dx^2} \).
- Determine the highest power of this highest order derivative. The power is 2.
The degree is 2.
| Characteristic | Value |
|---|---|
| Highest Order Derivative | \( \frac{d^2y}{dx^2} \) |
| Order | 2 |
| Equation after removing radical | \( 9 \frac{dy}{dx} = (y^2 + e^x - \frac{d^2y}{dx^2})^2 \) |
| Highest power of Highest Order Derivative (\( \frac{d^2y}{dx^2} \)) | 2 |
| Degree | 2 |
Revision Table: Differential Equation Concepts
| Concept | Definition | Example |
|---|---|---|
| Differential Equation | An equation involving derivatives of one or more dependent variables with respect to one or more independent variables. | \( \frac{dy}{dx} = xy \) |
| Order of a Differential Equation | The order of the highest derivative present in the differential equation. | For \( \frac{d^2y}{dx^2} + y = 0 \), Order is 2. |
| Degree of a Differential Equation | The highest power of the highest order derivative after the equation is made free of radicals and fractions in derivatives. | For \( (\frac{dy}{dx})^3 + (\frac{d^2y}{dx^2})^2 = 0 \), Order is 2, Degree is 2. |
| Linear Differential Equation | A differential equation where the dependent variable and its derivatives appear only in the first degree, and there are no products of the dependent variable and/or its derivatives. | \( \frac{dy}{dx} + P(x)y = Q(x) \) |
| Non-linear Differential Equation | A differential equation that is not linear. | \( (\frac{dy}{dx})^2 + y = 0 \) |
Additional Information: Properties of Differential Equations
Understanding the order and degree is fundamental to classifying and solving differential equations. Here are some related points:
- Polynomial in Derivatives: The concept of degree applies only when the differential equation can be written as a polynomial in the derivatives. If it contains terms like \( \sin(\frac{dy}{dx}) \) or \( e^{\frac{d^2y}{dx^2}} \), the degree is undefined.
- Simplification is Key: Always simplify the equation to remove radicals and fractions from the derivatives before determining the degree. This often involves algebraic steps like squaring or raising to a power.
- Order vs. Degree: The order depends only on the highest derivative's rank (\( \frac{dy}{dx}, \frac{d^2y}{dx^2} \), etc.), while the degree depends on the power of that highest ranked derivative *after simplification*.
- Importance: The classification by order and degree helps in selecting appropriate methods for solving differential equations.
Similar Questions
General solution of the differential equation \( \frac{2y dx - 3x dy}{y} = 0 \) is (c is an arbitrary constant):
The number of arbitrary constants in a particular solution of a differential equation of 4th order is:
The integrating factor of the differential equation:
\[ x \frac{dy}{dx} - 2y = x^3 \]
is:
Important Questions from Differential Equations
Anubhav spent 14% of his income on electricity bills, 28% on rent and 18% on shopping. If 4/5 of the remaining amount is ₹ 5120, how much did he spend on electricity bills?
Arun's speed of swimming in still water is 5 km/hr. He swims between two points in a river and returns back to the same starting point. He took 20 minutes more to cover the distance upstream than downstream. If the speed of the stream is 2 km/hr, then the distance between the two points is :
If \( f(x) = 2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \), then \( f(x) \) is:
Match List-I with List-II:
List-I List-II (A) Integrating factor of \( xdy - (y + 2x^2)dx = 0 \) (I) \( \frac{1}{x} \) (B) Integrating factor of \( (2x^2 - 3y)dx = xdy \) (II) \( x \) (C) Integrating factor of \( (2y + 3x^2)dx + xdy = 0 \) (III) \( x^2 \) (D) Integrating factor of \( 2xdy + (3x^3 + 2y)dx = 0 \) (IV) \( x^3 \) Choose the correct answer from the options given below:
If t = e2x and y = loge(t2), then d2y/dx2 is :
