OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Exercise 17 (A)

State the Order and the Degree of the Following Differential Equations:

Question 1. \( \frac { dy }{ dx } = \sin x \)
Answer: The given differential equation is \( \frac { dy }{ dx } = \sin x \). The highest order derivative in this equation is \( \frac { dy }{ dx } \). Its order is 1 because it's the first derivative. The power of this highest derivative is also 1. Therefore, the order of the differential equation is 1, and its degree is 1. Differential equations are fundamental in describing how quantities change relative to one another.
In simple words: The largest derivative here is \( \frac { dy }{ dx } \), which has an order of 1. The power of this largest derivative is also 1. So, the order is 1 and the degree is 1.

🎯 Exam Tip: To find the degree, the differential equation must be a polynomial in its derivatives. The order is determined by the highest derivative present, and the degree is the power of that highest derivative.

Question 2. \( x^2 \left(\frac { dy }{ dx }\right)^2 + 2y^2x = 0 \)
Answer: The given differential equation is \( x^2 \left(\frac { dy }{ dx }\right)^2 + 2y^2x = 0 \). In this equation, the highest order derivative present is \( \frac { dy }{ dx } \). This is a first-order derivative, so the order of the equation is 1. The power of this highest derivative, \( \left(\frac { dy }{ dx }\right)^2 \), is 2. Thus, the degree of the differential equation is 2. The degree indicates the highest power of the highest derivative once the equation is made free from fractions and radicals with respect to the derivatives.
In simple words: The largest derivative is \( \frac { dy }{ dx } \), which means the order is 1. The power of this largest derivative is 2. So, the order is 1 and the degree is 2.

🎯 Exam Tip: Remember to identify the highest order derivative first to find the order. Then, find its power to determine the degree, ensuring the equation is in polynomial form.

Question 3. \( \frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2 + x = 0 \)
Answer: The given differential equation is \( \frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2 + x = 0 \). The highest order derivative in this equation is \( \frac{d^2 y}{d x^2} \). Since it is a second-order derivative, the order of the differential equation is 2. The power of this highest order derivative \( \frac{d^2 y}{d x^2} \) is 1. Hence, the degree of the differential equation is 1. The order and degree are always positive integers, providing key information about the complexity of the equation.
In simple words: The highest derivative is \( \frac{d^2 y}{d x^2} \), which has an order of 2. The power of this highest derivative is 1. So, the order is 2 and the degree is 1.

🎯 Exam Tip: The order is the highest order of derivative present. The degree is the power of that highest order derivative, provided the equation is a polynomial in its derivatives.

Question 4. \( \left(\frac{d^2 y}{d x^2}\right)^2+\frac{d y}{dx} - xy = 0 \)
Answer: The given differential equation is \( \left(\frac{d^2 y}{d x^2}\right)^2+\frac{d y}{dx} - xy = 0 \). The highest derivative in this equation is \( \frac{d^2 y}{d x^2} \), which is a second-order derivative. Therefore, the order of the differential equation is 2. The power of this highest derivative term, \( \left(\frac{d^2 y}{d x^2}\right)^2 \), is 2. So, the degree of the differential equation is 2. Both order and degree are important properties of differential equations, influencing their solution methods.
In simple words: The highest derivative is \( \frac{d^2 y}{d x^2} \), which means its order is 2. The power on this highest derivative is 2. So, the order is 2 and the degree is 2.

🎯 Exam Tip: Make sure you correctly identify the highest order derivative before looking at its power to avoid mistakes in determining the degree.

Question 5. \( \frac{d^3 y}{d x^3}-5 \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^4 - 5x = 0 \)
Answer: The given differential equation is \( \frac{d^3 y}{d x^3}-5 \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^4 - 5x = 0 \). The highest order derivative present in this equation is \( \frac{d^3 y}{d x^3} \). This is a third-order derivative, so the order of the differential equation is 3. The power of this highest order derivative \( \frac{d^3 y}{d x^3} \) is 1. Therefore, the degree of the differential equation is 1. The order tells us the highest level of differentiation, while the degree tells us about its power in the polynomial form.
In simple words: The biggest derivative is \( \frac{d^3 y}{d x^3} \), which means its order is 3. This derivative has a power of 1. So, the order is 3 and the degree is 1.

🎯 Exam Tip: Always pick the highest derivative for determining the order. The degree comes from the power of *that specific* highest derivative, not any other.

Question 6. \( y = x \frac{d y}{d x}+\frac{a}{\frac{d y}{d x}} \)
Answer: The given differential equation is \( y = x \frac{d y}{d x}+\frac{a}{\frac{d y}{d x}} \). To find the degree, we first need to make this equation a polynomial in its derivatives by removing the fraction. We can do this by multiplying the entire equation by \( \frac{dy}{dx} \):
\( y \left(\frac { dy }{ dx }\right) = x \left(\frac { dy }{ dx }\right)^2 + a \)
Rearranging this gives a polynomial form:
\( x \left(\frac { dy }{ dx }\right)^2 - y \left(\frac { dy }{ dx }\right) + a = 0 \)
The highest order derivative in this modified equation is \( \frac { dy }{ dx } \). Its order is 1. The power of this highest derivative, \( \left(\frac { dy }{ dx }\right)^2 \), is 2. Therefore, the order of the differential equation is 1, and its degree is 2. Clearing fractions is a crucial step before determining the degree.
In simple words: First, rewrite the equation so there are no fractions with derivatives. It becomes \( x \left(\frac { dy }{ dx }\right)^2 - y \left(\frac { dy }{ dx }\right) + a = 0 \). The biggest derivative is \( \frac { dy }{ dx } \), which is order 1. The power of this derivative is 2. So, the order is 1 and the degree is 2.

🎯 Exam Tip: If the differential equation has derivatives in the denominator, multiply through by the derivative to make it a polynomial in the derivatives before determining the degree.

Question 7. \( (\sqrt{a+x})\left(\frac{d y}{d x}\right) + x = 0 \)
Answer: The given differential equation is \( (\sqrt{a+x})\left(\frac{d y}{d x}\right) + x = 0 \). The highest order derivative present in this equation is \( \frac{d y}{d x} \). Since this is a first-order derivative, the order of the differential equation is 1. The power of this highest order derivative \( \frac{d y}{d x} \) is 1. Thus, the degree of the differential equation is 1. Variables like \( x \) or \( y \) can appear under a radical, but derivatives themselves must not for the degree to be well-defined as a positive integer.
In simple words: The highest derivative is \( \frac{d y}{d x} \), so the order is 1. The power of this derivative is also 1. So, the order is 1 and the degree is 1.

🎯 Exam Tip: Be careful not to confuse square roots of variables (like \( \sqrt{a+x} \)) with square roots of derivatives. The latter can affect the degree.

Question 8. \( x \sqrt{1-y^2} d x+y \sqrt{1-x^2} d y = 0 \)
Answer: The given differential equation is \( x \sqrt{1-y^2} d x+y \sqrt{1-x^2} d y = 0 \). We can rewrite this equation to explicitly show the derivative \( \frac{dy}{dx} \).
\( y \sqrt{1-x^2} d y = -x \sqrt{1-y^2} d x \)
\( \frac{dy}{dx} = \frac{-x \sqrt{1-y^2}}{y \sqrt{1-x^2}} \)
In this form, the highest order derivative present is \( \frac{dy}{dx} \), which means its order is 1. The power of this highest derivative is 1. Therefore, the order of the differential equation is 1, and its degree is 1. This type of equation is often called a variable separable differential equation, useful in many physics problems.
In simple words: First, rearrange the equation to find \( \frac{dy}{dx} \). It will be \( \frac{dy}{dx} = \frac{-x \sqrt{1-y^2}}{y \sqrt{1-x^2}} \). The highest derivative is \( \frac{dy}{dx} \), so the order is 1. Its power is also 1. So, the order is 1 and the degree is 1.

🎯 Exam Tip: If the differential equation is not explicitly given in terms of derivatives, rearrange it first to identify the highest order derivative and its power correctly.

Question 9. \( \left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}=5 \frac{d^2 y}{d x^2} \)
Answer: The given differential equation is \( \left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}=5 \frac{d^2 y}{d x^2} \). To find the degree, we must first make the equation a polynomial in its derivatives by removing the fractional power. We do this by squaring both sides:
\( \left(\left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}\right)^2 = \left(5 \frac{d^2 y}{d x^2}\right)^2 \)
\( \left[1+\left(\frac{d y}{d x}\right)^2\right]^3 = 25\left(\frac{d^2 y}{d x^2}\right)^2 \)
Now, the highest order derivative in this modified equation is \( \frac{d^2 y}{d x^2} \). Its order is 2. The power of this highest derivative, \( \left(\frac{d^2 y}{d x^2}\right)^2 \), is 2. So, the order of the differential equation is 2, and its degree is 2. Removing fractional exponents is essential for determining the degree accurately.
In simple words: First, square both sides to remove the \( \frac{3}{2} \) power. This gives \( \left[1+\left(\frac{d y}{d x}\right)^2\right]^3 = 25\left(\frac{d^2 y}{d x^2}\right)^2 \). The highest derivative is \( \frac{d^2 y}{d x^2} \), so its order is 2. The power on this derivative is 2. So, the order is 2 and the degree is 2.

🎯 Exam Tip: Always ensure the differential equation is free from radicals and fractional powers of its derivatives before determining its degree.

Question 10. \( y = x \frac{d y}{dx}+a \sqrt{1+\left(\frac{d y}{d x}\right)^2} \)
Answer: The given differential equation is \( y = x \frac{d y}{dx}+a \sqrt{1+\left(\frac{d y}{d x}\right)^2} \). To find the degree, we must remove the radical sign from the derivative. First, isolate the term with the radical:
\( y - x \frac{d y}{d x} = a \sqrt{1+\left(\frac{d y}{d x}\right)^2} \)
Next, square both sides of the equation:
\( \left(y - x \frac{d y}{d x}\right)^2 = \left(a \sqrt{1+\left(\frac{d y}{d x}\right)^2}\right)^2 \)
\( \left(y - x \frac{d y}{d x}\right)^2 = a^2 \left(1+\left(\frac{d y}{d x}\right)^2\right) \)
Expanding this gives a polynomial in the derivatives. The highest order derivative in this equation is \( \frac{d y}{d x} \). Its order is 1. The highest power of this derivative, after expansion, will be 2 (from \( \left(\frac{d y}{d x}\right)^2 \) on both sides). Therefore, the order of the differential equation is 1, and its degree is 2. Isolating the radical and squaring is a standard procedure in such cases.
In simple words: First, move \( x \frac{d y}{d x} \) to the left side: \( y - x \frac{d y}{d x} = a \sqrt{1+\left(\frac{d y}{d x}\right)^2} \). Then, square both sides to get rid of the square root. This gives \( \left(y - x \frac{d y}{d x}\right)^2 = a^2 \left(1+\left(\frac{d y}{d x}\right)^2\right) \). The biggest derivative is \( \frac{d y}{d x} \), so its order is 1. The highest power of this derivative is 2. So, the order is 1 and the degree is 2.

🎯 Exam Tip: Always isolate any radical containing derivatives on one side of the equation before squaring to simplify and determine the degree correctly.

Question 11. \( \left(\frac{d^2 y}{d x^2}\right)^2=\left(\frac{d y}{d x}\right)^2 \)
Answer: The given differential equation is \( \left(\frac{d^2 y}{d x^2}\right)^2=\left(\frac{d y}{d x}\right)^2 \). The highest order derivative present in this equation is \( \frac{d^2 y}{d x^2} \). Since this is a second-order derivative, the order of the differential equation is 2. The power of this highest derivative, \( \left(\frac{d^2 y}{d x^2}\right)^2 \), is 2. Therefore, the degree of the differential equation is 2. In this specific case, the degree is simply the exponent of the highest derivative, even if other derivatives have higher exponents.
In simple words: The biggest derivative is \( \frac{d^2 y}{d x^2} \), so its order is 2. The power on this derivative is 2. So, the order is 2 and the degree is 2.

🎯 Exam Tip: Carefully identify the highest order derivative, then find its exponent to determine the degree. Do not be distracted by powers of lower-order derivatives.

Question 12. \( \frac{d y}{d x}=\frac{x}{d y / d x} \)
Answer: The given differential equation is \( \frac{d y}{d x}=\frac{x}{d y / d x} \). To make it a polynomial in its derivatives, we can multiply both sides by \( \frac{dy}{dx} \):
\( \left(\frac{d y}{d x}\right) \times \left(\frac{d y}{d x}\right) = x \)
\( \left(\frac{d y}{d x}\right)^2 = x \)
In this simplified form, the highest order derivative present is \( \frac{d y}{d x} \). Its order is 1. The power of this highest derivative, \( \left(\frac{d y}{d x}\right)^2 \), is 2. Therefore, the order of the differential equation is 1, and its degree is 2. Always simplify the equation to its polynomial form before determining the degree.
In simple words: Multiply both sides by \( \frac{dy}{dx} \) to get \( \left(\frac{d y}{d x}\right)^2 = x \). The biggest derivative is \( \frac{d y}{d x} \), so its order is 1. The power on this derivative is 2. So, the order is 1 and the degree is 2.

🎯 Exam Tip: Ensure the differential equation is free of fractions where derivatives appear in the denominator. Multiply to clear them before finding the degree.