Question:
If $m$ is order and $n$ is degree of the differential equation $y = x\frac{dy}{dx} + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$, then the value of $m + n$ is
If $m$ is order and $n$ is degree of the differential equation $y = x\frac{dy}{dx} + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$, then the value of $m + n$ is
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Whenever you see a differential equation with a derivative inside a square root or fraction, your very first step must always be algebraically isolating and eliminating that radical by squaring. You cannot determine the true degree otherwise.
Updated On: Jun 4, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
We need to determine the order ($m$) and degree ($n$) of the given differential equation (which is in Clairaut's form), and then find their sum.
Step 2: Key Formula or Approach:
The order of a differential equation is the highest derivative present. The degree is the highest power of that derivative, but only after the equation has been cleared of any fractional powers or radicals involving the derivatives.
Step 3: Detailed Explanation:
The given differential equation is:
$$y = x\left(\frac{dy}{dx}\right) + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$$ To find the degree, we must clear the radical. Isolate the square root term on one side:
$$y - x\left(\frac{dy}{dx}\right) = \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$$ Square both sides of the equation:
$$\left(y - x\left(\frac{dy}{dx}\right)\right)^2 = a^2 \left(\frac{dy}{dx}\right)^2 - b^2$$ Expand the left side:
$$y^2 - 2xy\left(\frac{dy}{dx}\right) + x^2\left(\frac{dy}{dx}\right)^2 = a^2 \left(\frac{dy}{dx}\right)^2 - b^2$$ Group the derivative terms to see the polynomial form clearly:
$$(x^2 - a^2)\left(\frac{dy}{dx}\right)^2 - 2xy\left(\frac{dy}{dx}\right) + (y^2 + b^2) = 0$$ Now, analyze this simplified equation:
1. The highest order derivative present is $\frac{dy}{dx}$, which is a first-order derivative. Therefore, the order $m = 1$.
2. The highest power raised to this first-order derivative is 2. Therefore, the degree $n = 2$.
The question asks for the sum of $m$ and $n$:
$$m + n = 1 + 2 = 3$$
Step 4: Final Answer:
The value of $m + n$ is 3, matching option (B).
We need to determine the order ($m$) and degree ($n$) of the given differential equation (which is in Clairaut's form), and then find their sum.
Step 2: Key Formula or Approach:
The order of a differential equation is the highest derivative present. The degree is the highest power of that derivative, but only after the equation has been cleared of any fractional powers or radicals involving the derivatives.
Step 3: Detailed Explanation:
The given differential equation is:
$$y = x\left(\frac{dy}{dx}\right) + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$$ To find the degree, we must clear the radical. Isolate the square root term on one side:
$$y - x\left(\frac{dy}{dx}\right) = \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 - b^2}$$ Square both sides of the equation:
$$\left(y - x\left(\frac{dy}{dx}\right)\right)^2 = a^2 \left(\frac{dy}{dx}\right)^2 - b^2$$ Expand the left side:
$$y^2 - 2xy\left(\frac{dy}{dx}\right) + x^2\left(\frac{dy}{dx}\right)^2 = a^2 \left(\frac{dy}{dx}\right)^2 - b^2$$ Group the derivative terms to see the polynomial form clearly:
$$(x^2 - a^2)\left(\frac{dy}{dx}\right)^2 - 2xy\left(\frac{dy}{dx}\right) + (y^2 + b^2) = 0$$ Now, analyze this simplified equation:
1. The highest order derivative present is $\frac{dy}{dx}$, which is a first-order derivative. Therefore, the order $m = 1$.
2. The highest power raised to this first-order derivative is 2. Therefore, the degree $n = 2$.
The question asks for the sum of $m$ and $n$:
$$m + n = 1 + 2 = 3$$
Step 4: Final Answer:
The value of $m + n$ is 3, matching option (B).
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