PSEB 12th Class Maths Solutions Chapter 9 Differential Equations Ex 9.1

Punjab State Board PSEB 12th Class Maths Book Solutions Chapter Differential Equations Ex 9.1 Textbook Exercise Questions and Answers.

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

Direction (1 – 10): Determine order and degree (if defined) of differential equation.

Question 1.
\(\frac{d^{4} y}{d x^{4}}\) + sin(y””) = 0.
Solution.
\(\frac{d^{4} y}{d x^{4}}\) + sin(y””) = 0
⇒ y”” + sin (y””) = 0
The highest order derivative present in the differential equation is y””, therefore, its order is 4.
The given differential equation is not a polynomial equation in its derivatives.
Hence, its degree is not defined.

Question 2.
y’ + 5y = 0.
Solution.
The given differential equation is y’ + 5y = 0
The highest order derivative present in the differential equation is y’.
Therefore, its order is 1.
It is a polynomial equation in y’. The highest power raised to y’ is 1.
Hence, its degree is 1.

Question 3.
\(\left(\frac{d s}{d t}\right)^{4}+3 \frac{d^{2} s}{d t^{2}}\) = 0
Solution.
\(\left(\frac{d s}{d t}\right)^{4}+3 \frac{d^{2} s}{d t^{2}}\) = 0
The highest order derivative present in the given differential equation is \(\frac{d^{2} s}{d t^{2}}\), therefore, its order is 2.
It is a polynomial equation in \(\frac{d^{2} s}{d t^{2}}\) and \(\frac{d s}{d t}\).
The highest power raised to \(\frac{d^{2} s}{d t^{2}}\) is 1.
Hence, 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.
Solution.
\(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}\) + cos \(\left(\frac{d y}{d x}\right)\) = 0.
The highest order derivative present in the given differential equation is \(\frac{d^{2} y}{d x^{2}}\). Therefore, its order is 2.
The given differential equation is not a polynomial equation in its derivatives.
Hence, its degree is not defined.

Question 5.
\(\frac{d^{2} y}{d x^{2}}\) = cos 3x + sin 3x
Solution.
\(\frac{d^{2} y}{d x^{2}}\) = cos 3x + sin 3x

⇒ \(\frac{d^{2} y}{d x^{2}}\) – cos 3x – sin 3x = 0
The highest order derivative present in the differential equation is \(\frac{d^{2} y}{d x^{2}}\).
Therefore its order is 2.
It is a polynomial equation in \(\frac{d^{2} y}{d x^{2}}\) and the p[ower raised to \(\frac{d^{2} y}{d x^{2}}\) is 1.
Hence, its degree is 1.

Question 6.
(y”’)² + (y”)³ + (y’)⁴ + y⁵ = 0.
Solution.
(y”’)² + (y”)³ + (y’)⁴ + y⁵ = 0
The highest order derivative present in the differential equation is y”’.
Therefore, its order is 3.
The given differential equation is a polynomial equation in y”’, y”, and y’.
The highest power raised to y” is 2.
Hence, its degree is 2.

Question 7.
y”’ + 2y” + y’= 0.
Solution.
y”’ + 2y” + y’ = 0
The highest order derivative present in the differential equation is y”’.
Therefore, its order is 3.
It is polynomial equation in y”‘, y” and y’.
The highest power raised to y”‘ is 1.
Hence, its degree is 1.

Question 8.
y’ + y = e^x.
Solution.
y’ + y = e^x
⇒ y’ + y – e^x = 0
The highest order derivative present in the differential equation is y’.
Therefore, its order is 1.
The given differential equation is a polynomial equation in y’ and the highest power raised to y is 1.
Hence, its degree is 1.

Question 9.
y” + (y’)² + 2y = 0.
Solution.
y” + (y’)² + 2y = 0
The highest order derivative present in the differential equation is y”.
Therefore, its order is 2.
The given differential equation is a polynomial equation in y” and y’ and the highest power raised to y” is 1.
Hence, its degree is 1.

Question 10.
y” + 2y’ + sin y = 0.
Solution.
y” + 2y’ + sin y = 0
The highest order derivative present in the differential equation is y”.
Therefore, its order is 2.
This is a polynomial equation in y” and y’ and the highest power raised to y” is 1.
Hence, its degree is 1.

Direction (11 – 12): Choose the correct answer.

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) None of these
Solution.
\(\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}\) + sin \(\left(\frac{d y}{d x}\right)\) + 1 = 0
The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.
Hence, the correct answer is (D).

Question 12.
The order of the differential equation 2x² \(\frac{d^{2} y}{d x^{2}}\) – 3 \(\frac{d y}{d x}\) + y = 0
(A) 2
(B) 1
(C) zero
(D) None of these
Solution.
2x² \(\frac{d^{2} y}{d x^{2}}\) – 3 \(\frac{d y}{d x}\) + y = 0

The highest order derivative present in the given differential equation is \(\frac{d^{2} y}{d x^{2}}\)
Therefore, its order is 2.
Hence, the correct answer is (A).