Exercise 9.1 Class 12 Maths Differential Equations

NCERT Solutions Class 12 Maths Chapter 9 Ex 9.1 | Differential Equations

💡 Definitions

Order: The order of the highest order derivative appearing in the equation.

Degree: The power of the highest order derivative, when the equation is a polynomial in derivatives.

[Image of order and degree of differential equation examples]

Note: If the equation involves $\sin(y’), \cos(y’), e^{y’}, \log(y’)$, etc., the Degree is Not Defined.

Questions 01 — 03

Find Order and Degree.

1. $\frac{d^4y}{dx^4} + \sin(y”’) = 0$

Highest derivative is $\frac{d^4y}{dx^4}$ (Order 4).
The term $\sin(y”’)$ makes it non-polynomial in derivatives.

Order: 4, Degree: Not Defined

2. $y’ + 5y = 0$

Highest derivative is $y’$ (Order 1).
Power of $y’$ is 1.

Order: 1, Degree: 1

3. $(\frac{ds}{dt})^4 + 3s\frac{d^2s}{dt^2} = 0$

Highest derivative is $\frac{d^2s}{dt^2}$ (Order 2).
Power of $\frac{d^2s}{dt^2}$ is 1. (Ignore power 4 of first derivative).

Order: 2, Degree: 1

Questions 04 — 06

Tricky Cases.

4. $(\frac{d^2y}{dx^2})^2 + \cos(\frac{dy}{dx}) = 0$

Highest derivative is $\frac{d^2y}{dx^2}$ (Order 2).
Contains $\cos(\frac{dy}{dx})$, so it’s not a polynomial in derivatives.

Order: 2, Degree: Not Defined

5. $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$

Highest derivative is $\frac{d^2y}{dx^2}$ (Order 2).
Power is 1. (Trig functions of $x$ do not affect degree).

Order: 2, Degree: 1

6. $(y”’)^2 + (y”)^3 + (y’)^4 + y^5 = 0$

Highest derivative is $y”’$ (Order 3).
Power of $y”’$ is 2.

Order: 3, Degree: 2

Questions 07 — 10

Standard Forms.

7. $y”’ + 2y” + y’ = 0$

Order: 3, Degree: 1

8. $y’ + y = e^x$

Order: 1, Degree: 1

9. $y” + (y’)^2 + 2y = 0$

Highest derivative $y”$ (Order 2). Power is 1.

Order: 2, Degree: 1

10. $y” + 2y’ + \sin y = 0$

Highest derivative $y”$ (Order 2).
Note: $\sin y$ is allowed. Only trig functions of derivatives (like $\sin y’$) make degree undefined.

Order: 2, Degree: 1

Questions 11 — 12

Multiple Choice Questions.

11. Degree of $(\frac{d^2y}{dx^2})^3 + (\frac{dy}{dx})^2 + \sin(\frac{dy}{dx}) + 1 = 0$

Highest derivative is $\frac{d^2y}{dx^2}$.
However, due to the presence of $\sin(\frac{dy}{dx})$, the equation is not a polynomial in its derivatives.

Correct Option: (D) Not defined

12. Order of $2x^2 \frac{d^2y}{dx^2} – 3\frac{dy}{dx} + y = 0$

The highest order derivative present is $\frac{d^2y}{dx^2}$.

Correct Option: (A) 2