Order and degree of a differential equation \(\frac{d^2y}{dx^2} = \left\{ y + \left(\frac{dy}{dx}\right)^2 \right\}^{1/4}\) are
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- 4 and 2
- 1 and 2
- 1 and 4
- 2 and 4
The Correct Option is D
Solution and Explanation
The given differential equation is:
\(\frac{d^2y}{dx^2} = \left\{ y + \left(\frac{dy}{dx}\right)^2 \right\}^{1/4}\)
To find the order and degree of this differential equation, we need to understand the definitions:
- The order of a differential equation is the highest order of the derivative present in the equation.
- The degree of a differential equation is the power of the highest order derivative after the equation has been made free from radicals and fractions with respect to the derivatives.
Based on this, let's determine the order first:
The highest order derivative in the equation is \(\frac{d^2y}{dx^2}\), so the order of the differential equation is 2.
Next, to find the degree, we must ensure there are no fractional or radical powers involving the dependent variable or its derivatives.
The equation is:
\(\frac{d^2y}{dx^2} = \{y + \left(\frac{dy}{dx}\right)^2\}^{1/4}\)
To eliminate the fractional power, we raise both sides of the equation to the power of 4:
\(\left( \frac{d^2y}{dx^2} \right)^4 = y + \left(\frac{dy}{dx}\right)^2\)
Now, the degree of the equation is the power of \(\frac{d^2y}{dx^2}\), which is 4.
Therefore, the order and degree of the given differential equation are:
- Order: 2
- Degree: 4
Thus, the correct option is: 2 and 4.
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