RBSE Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.1

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.1 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 9 Differential Equations Ex 9.1

Question 1.
\(\left(\frac{d^2 y}{d x^2}\right)^2\) + sin (y''') = 0.
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^4 y}{d x^4}\), so its order is 4. The given differential equation is not a polynomial equation in its derivative and so its degree is not defined.

Question 2.
y’+ 5y = 0 or \(\frac{d y}{d x}\) + 5y = 0
Answer:
The highest order derivative present in the differential equation is \(\frac{d y}{d x}\), so its order is 1. It is a polynomial equation in \(\frac{d y}{d x}\) and the highest power raised to \(\frac{d y}{d x}\) is 1, so degree is 1.

Question 3.
\(\left(\frac{d s}{d t}\right)^4\) + 3s\(\frac{d^2 s}{d t^2}\) = 0
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^2 s}{d t^2}\), so its order is 2. It is a polynomial equation in \(\frac{d^2 s}{d t^2}\) and \(\frac{d s}{d t}\) and the highest power raised to \(\frac{d^2 s}{d t^2}\) is 1, so its degree is 1.

Question 4.
\(\left(\frac{d^2 y}{d x^2}\right)^2\) + cos \(\left(\frac{d y}{d x}\right)\) = 0
Answer:
The highest order derivative present in the given differential equation is so its order is 2. The given differential equation is \(\frac{d^2 y}{d x^2}\), so its order is 2. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined.

Question 5.
\(\frac{d^2 y}{d x^2}\) = cos 3x + sin 3x
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^2 y}{d x^2}\), so its order is 2. It is a polynomial equation in \(\frac{d^2 y}{d x^2}\) and the highest power raised to \(\frac{d^2 y}{d x^2}\) is 1, so its degree is 1.

Question 6.
(y''')² + (y'')³ + (y')⁴ + y⁵ = 0
or \(\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^4\) + y⁵ = 0
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^3 y}{d x^3}\), so its order is 3. It is a polynomial equation in \(\frac{d^3 y}{d x^3}\), \(\frac{d^2 y}{d x^2}\) and \(\frac{d y}{d x}\) and the highest power raised to \(\frac{d^3 y}{d x^3}\) is 2, so its degree is 2.

Question 7.
y''' + 2y'' + y’ = 0 or \(\frac{d^3 y}{d x^3}+2 \frac{d^2 y}{d x^2}+\frac{d y}{d x}\) = 0
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^3 y}{d x^3}\), so its order is 3. It is a polynomial equation in \(\frac{d^3 y}{d x^3}\), \(\frac{d^2 y}{d x^2}\) and \(\frac{d y}{d x}\) and the highest power raised to \(\frac{d^3 y}{d x^3}\) is 1, so its degree is 1.

Question 8.
y' + y = e^x or \(\frac{d y}{d x}\) + y = e^x.
Answer:
The highest order derivative present in the given differential equation is \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), so its order is 1. It is a polynomial equation in \(\frac{d y}{d x}\) and the power raised to \frac{d y}{d x} is 1, so its degree is 1.

Question 9.
y'' + (y')² + 2y = 0 or \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2\) + 2y = 0
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^2 y}{d x^2}\) so its order is 2. It is a polynomial equation in \(\frac{d^3 y}{d x^3}, \frac{d^2 y}{d x^2}\) and \(\frac{d y}{d x}\) the highest power raised to \(\frac{d^2 y}{d x^2}\) is 1, so its degree is 1.

Question 10.
y'' + 2y' + sin y = 0
or \(\frac{d^2 y}{d x^2}\) + 2\(\frac{d y}{d x}\) + sin y = 0
Answer:
The highest order derivative present in the given differential equation is \(\frac{d^2 y}{d x^2}\), so its order is 2. It is a polynomial equation in \(\frac{d^3 y}{d x^3}, \frac{d^2 y}{d x^2}\) and \(\frac{d y}{d x}\) and the highest power raised to \(\frac{d^2 y}{d x^2}\) is 2, so its degree is 1.

Question 11.
The degree of the differential equation
\(\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2 + sin \left(\frac{d y}{d x}\right)\) + 1 = 0 is
(A) 3
(B) 2
(C) 1
(D) not defined
Answer:
The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined,
Hence, (D) is the correct answer.

Question 12.
The order of the differential equation
2x²\(\frac{d^2 y}{d x^2}\) - 3\(\frac{d y}{d x}\) + y = 0 is:
(A) 2
(B) 1
(C) 0
(D) not defined
Answer:
The highest order derivative present in the differential equation is \(\frac{d^2 y}{d x^2}\), so its order is 2.
Hence, (A) is the correct answer.