GSEB Class 12 Maths Solutions Chapter 9 Differential Equations Exercise 9.1

Determine orders and degrees (if defined) of the differential equations given in questions 1 to 10:

Question 1. \( \frac{d^4 y}{d x^4} + \sin(y''') = 0 \)
Answer: The highest order derivative in this equation is \( \frac{d^4 y}{d x^4} \). Therefore, the order of the differential equation is 4. The degree of this differential equation is not defined because the \( \sin \) function contains a derivative (\( y''' \)). For a degree to be defined, the differential equation must be a polynomial in its derivatives.
In simple words: The highest derivative gives you the order, which is 4 here. The degree isn't defined because a derivative is inside a sine function, meaning it's not a simple polynomial equation.

Exam Tip: Remember that the degree of a differential equation is only defined if it can be expressed as a polynomial in its derivatives. If any derivative is within a trigonometric, exponential, or logarithmic function, the degree is undefined.

Question 2. \( y' + 5y = 0 \)
Answer: The highest order derivative in this equation is \( y' \) (which is \( \frac{dy}{dx} \)). Thus, the order of the differential equation is 1. Since \( y' \) is raised to the power of 1, the degree of the differential equation is 1.
In simple words: The biggest derivative is just the first one, so the order is 1. That first derivative has a power of 1, so the degree is also 1.

Exam Tip: For simple linear differential equations like this, both the order and degree are often 1, representing the simplest form of such equations.

Question 3. \( (\frac{d s}{d t})^4 + 3(\frac{d^2 s}{d t^2}) = 0 \)
Answer: The highest order derivative present in this equation is \( \frac{d^2 s}{d t^2} \). Therefore, the order of the differential equation is 2. The power of this highest order derivative is 1, so the degree of the differential equation is 1.
In simple words: The second derivative is the highest one, making the order 2. Since that second derivative is only raised to the power of 1, the degree is also 1.

Exam Tip: Always look for the highest order derivative first to determine the order, then check its exponent to find the degree, provided the equation is a polynomial in derivatives.

Question 4. \( (\frac{d^2 y}{d x^2})^2 + \cos(\frac{dy}{dx}) = 0 \)
Answer: The highest order derivative in this equation is \( \frac{d^2 y}{d x^2} \). Consequently, the order of the differential equation is 2. However, because the \( \cos \) function contains a derivative (\( \frac{dy}{dx} \)), the equation is not a polynomial in its derivatives. Therefore, the degree of this differential equation is not defined.
In simple words: The biggest derivative is the second one, so the order is 2. But since a derivative is inside the cosine function, the degree cannot be defined for this equation.

Exam Tip: Be cautious with trigonometric functions or exponents involving derivatives; they often cause the degree of the differential equation to be undefined.

Question 5. \( \frac{d^2 y}{d x^2} = \cos 3x + \sin 3x \)
Answer: The highest order derivative in this equation is \( \frac{d^2 y}{d x^2} \). So, the order of the differential equation is 2. The highest order derivative is raised to the power of 1. Therefore, the degree of the differential equation is 1.
In simple words: The highest derivative here is the second one, making the order 2. That second derivative has a power of 1, so the degree is 1.

Exam Tip: Functions of the independent variable (like \( \cos 3x \)) do not affect the degree of the differential equation, only the derivatives themselves matter.

Question 6. \( (y''')^2 + (y'')^3 + (y')^4 + y^5 = 0 \)
Answer: The highest order derivative in this equation is \( y''' \) (the third derivative). Hence, the order of the differential equation is 3. The highest power to which the highest order derivative (\( y''' \)) is raised is 2. Thus, the degree of the differential equation is 2.
In simple words: The biggest derivative is the third one, so the order is 3. That third derivative is squared, making the degree 2.

Exam Tip: When determining the degree, always check the power of the *highest order* derivative, not necessarily the highest power of any derivative in the equation.

Question 7. \( y''' + 2y^2 + y = 0 \)
Answer: The highest order derivative in this equation is \( y''' \) (the third derivative). So, the order of the differential equation is 3. The highest order derivative (\( y''' \)) is raised to the power of 1. Therefore, the degree of the differential equation is 1.
In simple words: The third derivative is the highest one, giving an order of 3. Since that third derivative has a power of 1, the degree is also 1.

Exam Tip: Be careful not to confuse the powers of the dependent variable (\( y^2 \)) with the powers of its derivatives when finding the degree.

Question 8. \( y' + y = e^x \)
Answer: The highest order derivative in this equation is \( y' \) (the first derivative). Therefore, the order of the differential equation is 1. The highest order derivative (\( y' \)) is raised to the power of 1. Thus, the degree of the differential equation is 1.
In simple words: The highest derivative is the first one, making the order 1. Since that first derivative is only to the power of 1, the degree is 1 as well.

Exam Tip: The presence of non-derivative terms like \( e^x \) does not influence the calculation of order or degree; only the derivatives and their powers matter.

Question 9. \( y'' + (y')^2 + 2y = 0 \)
Answer: The highest order derivative in this equation is \( y'' \) (the second derivative). Hence, the order of the differential equation is 2. The highest order derivative (\( y'' \)) is raised to the power of 1. Therefore, the degree of the differential equation is 1.
In simple words: The second derivative is the highest, so the order is 2. The power of that second derivative is 1, meaning the degree is 1.

Exam Tip: Even if a lower-order derivative (like \( y' \)) has a higher power (\( (y')^2 \)), the degree is still determined by the power of the *highest* order derivative.

Question 10. \( y'' + 2y' + \sin y = 0 \)
Answer: The highest order derivative in this equation is \( y'' \) (the second derivative). Thus, the order of the differential equation is 2. The highest order derivative (\( y'' \)) is raised to the power of 1. Consequently, the degree of the differential equation is 1.
In simple words: The highest derivative is the second one, so the order is 2. That second derivative has a power of 1, making the degree 1. The \( \sin y \) term does not affect the order or degree.

Exam Tip: Only derivatives embedded within functions (like \( \sin(\frac{dy}{dx}) \)) affect the degree; terms like \( \sin y \) (where \( y \) is not a derivative) do not.

Choose the correct answers in the following questions 11 and 12:

Question 11. The degree of differential equation \( \left(\frac{d^{2} y}{d x^{2}}\right)³ + \left(\frac{1}{2}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0 \) is
(A) 1
(B) 2
(C) 1
(D) not defined
Answer: (D) not defined
In simple words: This equation has a derivative inside a sine function. Because of that, it's not a polynomial in terms of its derivatives, so the degree is not defined.

Exam Tip: The presence of a derivative within a transcendental function (like \( \sin \), \( \cos \), \( e \), or \( \log \)) immediately makes the degree of the differential equation undefined, even if other terms are polynomial.

Question 12. The order of differential equation \( 2x^2\frac{d^{2} y}{d x^{2}} – 3\frac{dy}{dx} + y = 0 \) is
(A) 1
(B) 1
(C) 0
(D) not defined
Answer: (D) not defined
The highest order derivative in the given differential equation is \( \frac{d^{2} y}{d x^{2}} \). This means the order of the differential equation is 2. Since '2' is not provided as an option among (A), (B), or (C), we select (D) not defined, indicating that the correct order is not listed in the given choices.
In simple words: The highest derivative is the second one, so the order of this equation should be 2. Since 2 is not an option, the correct answer is 'not defined' from the choices provided.

Exam Tip: Always correctly identify the highest derivative to determine the order. If the calculated order is not among the options, choose the "not defined" or "none of the options" choice if available.