RBSE Solutions Class 12 Maths Chapter 12 Differential Equation Exercise 12.1

Rajasthan Board RBSE Class 12 Maths Chapter 12 Differential Equation Ex 12.1

Find the Order and Degree of the Following Differential Equations:

Question 1. \( \frac { dy }{ dx} = \sin 2x + \cos 2x \)
Answer: The given differential equation is \( \frac { dy }{ dx} = \sin 2x + \cos 2x \). In this equation, the highest derivative is \( \frac{dy}{dx} \). The power of this highest derivative is 1. This means the differential equation has an order of 1 and a degree of 1. Identifying the highest derivative and its power is key to determining order and degree.
In simple words: The biggest derivative here is \( \frac{dy}{dx} \). Its power is one. So, the equation's order is 1, and its degree is 1.

🎯 Exam Tip: Always look for the highest order derivative in the equation first to determine the order, then find its power to get the degree.

Question 2. \( \frac {{d}^{2}y }{ { dx }^{ 2 } } = \sin x + \cos x \)
Answer: The given differential equation is \( \frac {{d}^{2}y }{ { dx }^{ 2 } } = \sin x + \cos x \). The highest derivative present in this equation is \( \frac{d^2y}{dx^2} \). The power of this highest derivative is 1. Therefore, the order of the differential equation is 2, and its degree is 1. The order indicates the number of times the function has been differentiated.
In simple words: The highest derivative in this equation is \( \frac{d^2y}{dx^2} \). Its power is one. So, the order is 2, and the degree is 1.

🎯 Exam Tip: Remember that the order refers to the highest derivative itself (like 1st, 2nd, or 3rd derivative), while the degree is the power of that highest derivative.

Question 3. \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = 0 \)
Answer: The given differential equation is \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = 0 \). The highest derivative in this equation is \( \frac{d^2y}{dx^2} \). The power of this highest derivative is 2. Thus, the order of the differential equation is 2, and its degree is 2. Both order and degree help classify differential equations.
In simple words: The biggest derivative is \( \frac{d^2y}{dx^2} \). Its power is two. So, the order is 2, and the degree is 2.

🎯 Exam Tip: Ensure the equation is expressed as a polynomial in its derivatives before determining the degree, especially if there are fractional powers or non-polynomial terms.

Question 4. \( \left(\frac{dy}{dx}\right)^3 + \frac{1}{\frac{dy}{dx}} = 2 \)
Answer: The given differential equation is \( \left(\frac{dy}{dx}\right)^3 + \frac{1}{\frac{dy}{dx}} = 2 \). To find the degree, we first need to make it a polynomial in terms of its derivatives.
\( \implies \left(\frac{dy}{dx}\right)^4 + 1 = 2 \left(\frac{dy}{dx}\right) \) The highest derivative in the polynomial form is \( \frac{dy}{dx} \). The power of this highest derivative is 4. Therefore, the order of the differential equation is 1, and its degree is 4. This step of clearing the denominator is crucial for correctly identifying the degree.
In simple words: First, rewrite the equation so there are no derivatives in the bottom of a fraction. After that, the biggest derivative is \( \frac{dy}{dx} \), and its power is four. So, the order is 1, and the degree is 4.

🎯 Exam Tip: If derivatives appear in fractions or under radicals, always manipulate the equation to clear these first so it becomes a polynomial in derivatives. Only then can you correctly determine the degree.

Question 5. \( a \frac{d^2y}{dx^2} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} \)
Answer: The given differential equation is \( a \frac{d^2y}{dx^2} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} \). To find the degree, we must remove the fractional power.
Squaring both sides:
\( \implies a^2 \left(\frac{d^2y}{dx^2}\right)^2 = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 \) In this polynomial form, the highest derivative is \( \frac{d^2y}{dx^2} \). The power of this highest derivative is 2. Thus, the order of the differential equation is 2, and its degree is 2. Squaring both sides helps make the equation a polynomial in its derivatives.
In simple words: We square both sides to get rid of the fraction power. Then, the biggest derivative is \( \frac{d^2y}{dx^2} \), and its power is two. So, the order is 2, and the degree is 2.

🎯 Exam Tip: When an equation has a derivative raised to a fractional power, isolate that term and raise both sides to an appropriate power to eliminate the fraction before determining the degree.

Question 6. \( x\,dx + y\,dy = 0 \)
Answer: The given differential equation is \( x\,dx + y\,dy = 0 \).
We can rewrite this by dividing by \( dx \):
\( \implies x + y \frac{dy}{dx} = 0 \) In this form, the highest derivative is \( \frac{dy}{dx} \). The power of this highest derivative is 1. Therefore, the order of the differential equation is 1, and its degree is 1. This shows how simple rearrangement can reveal the derivative.
In simple words: We change the equation to show \( \frac{dy}{dx} \). The biggest derivative is \( \frac{dy}{dx} \), and its power is one. So, the order is 1, and the degree is 1.

🎯 Exam Tip: Always express the differential equation in terms of derivatives like \( \frac{dy}{dx} \), \( \frac{d^2y}{dx^2} \), etc., before finding its order and degree.

Question 7. \( \left(\frac{d^2y}{dx^2}\right)^3 + y\left(\frac{dy}{dx}\right)^2 + y^5 = 0 \)
Answer: The given differential equation is \( \left(\frac{d^2y}{dx^2}\right)^3 + y\left(\frac{dy}{dx}\right)^2 + y^5 = 0 \). This equation is already in a polynomial form in terms of its derivatives. The highest derivative present is \( \frac{d^2y}{dx^2} \). The power of this highest derivative is 3. Thus, the order of the differential equation is 2, and its degree is 3. The coefficients of the derivative terms are \( y \) and constants, which is acceptable.
In simple words: The biggest derivative here is \( \frac{d^2y}{dx^2} \). Its power is three. So, the order is 2, and the degree is 3.

🎯 Exam Tip: When finding the degree, only consider the power of the *highest order* derivative. Powers of lower order derivatives do not affect the overall degree.

Question 8. \( x \frac{dy}{dx} + \frac{3}{\frac{dy}{dx}} = y^2 \)
Answer: The given differential equation is \( x \frac{dy}{dx} + \frac{3}{\frac{dy}{dx}} = y^2 \). To determine the degree, we first transform this into a polynomial in its derivatives.
Multiply the entire equation by \( \frac{dy}{dx} \):
\( \implies x \left(\frac{dy}{dx}\right)^2 + 3 = y^2 \frac{dy}{dx} \) In this polynomial form, the highest derivative is \( \frac{dy}{dx} \). The power of this highest derivative is 2. Therefore, the order of the differential equation is 1, and its degree is 2. This process ensures all derivatives are in numerator positions for degree determination.
In simple words: We multiply by \( \frac{dy}{dx} \) to get rid of the fraction. Then, the biggest derivative is \( \frac{dy}{dx} \), and its power is two. So, the order is 1, and the degree is 2.

🎯 Exam Tip: If a derivative appears in the denominator, multiply the entire equation by that derivative to clear the fraction and convert it into a polynomial form.