Exercise 9.1 Practice
Chapter 9: Differential Equations – NCERT Solutions
Q1
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Determine order and degree (if defined) of differential equation: $\frac{d^4y}{dx^4} + \sin(y''') = 0$
The highest order derivative present is $y''''$, so the order is 4.
The given differential equation is not a polynomial equation in its derivatives because of the term $\sin(y''')$. Therefore, its degree is not defined.
$\boxed{\text{Order: 4, Degree: Not defined}}$
Q2
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Determine order and degree (if defined) of differential equation: $y' + 5y = 0$
The highest order derivative present is $y'$, so the order is 1.
It is a polynomial equation in $y'$ and the highest power raised to $y'$ is 1. So, the degree is 1.
$\boxed{\text{Order: 1, Degree: 1}}$
Q3
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Determine order and degree (if defined) of differential equation: $(\frac{ds}{dt})^4 + 3s \frac{d^2s}{dt^2} = 0$
The highest order derivative present is $\frac{d^2s}{dt^2}$, so the order is 2.
The power of the highest order derivative is 1. So, the degree is 1.
$\boxed{\text{Order: 2, Degree: 1}}$
Q4
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Determine order and degree (if defined) of differential equation: $(\frac{d^2y}{dx^2})^2 + \cos(\frac{dy}{dx}) = 0$
The highest order derivative present is $\frac{d^2y}{dx^2}$, so the order is 2.
The equation involves $\cos(\frac{dy}{dx})$, so it is not a polynomial in derivatives. Degree is not defined.
$\boxed{\text{Order: 2, Degree: Not defined}}$
Q5
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Determine order and degree (if defined) of differential equation: $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$
The highest order derivative is $\frac{d^2y}{dx^2}$, so order is 2.
The power of the highest order derivative is 1. So, degree is 1.
$\boxed{\text{Order: 2, Degree: 1}}$
Q6
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Determine order and degree (if defined) of differential equation: $(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0$
Highest order derivative is $y'''$ (order 3).
The power of $y'''$ is 2. So, degree is 2.
$\boxed{\text{Order: 3, Degree: 2}}$
Q7
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Determine order and degree (if defined) of differential equation: $y''' + 2y'' + y' = 0$
Highest order derivative is $y'''$ (order 3).
Power of $y'''$ is 1. So, degree is 1.
$\boxed{\text{Order: 3, Degree: 1}}$
Q8
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Determine order and degree (if defined) of differential equation: $y' + y = e^x$
Highest order derivative is $y'$ (order 1).
Power of $y'$ is 1. So, degree is 1.
$\boxed{\text{Order: 1, Degree: 1}}$
Q9
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Determine order and degree (if defined) of differential equation: $y'' + (y')^2 + 2y = 0$
Highest order derivative is $y''$ (order 2).
Power of $y''$ is 1. So, degree is 1.
$\boxed{\text{Order: 2, Degree: 1}}$
Q10
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Determine order and degree (if defined) of differential equation: $y'' + 2y' + \sin y = 0$
Highest order derivative is $y''$ (order 2).
Power of $y''$ is 1. So, degree is 1. (Note: $\sin y$ is allowed, only derivatives must be polynomial).
$\boxed{\text{Order: 2, Degree: 1}}$
Q11
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The degree of the differential equation $(\frac{d^2y}{dx^2})^3 + (\frac{dy}{dx})^2 + \sin(\frac{dy}{dx}) + 1 = 0$ is:
(A) 3 (B) 2 (C) 1 (D) not defined
(A) 3 (B) 2 (C) 1 (D) not defined
The equation involves $\sin(\frac{dy}{dx})$, so it is not a polynomial in derivatives.
$\boxed{\text{(D) not defined}}$
Q12
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The order of the differential equation $2x^2 \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0$ is:
(A) 2 (B) 1 (C) 0 (D) not defined
(A) 2 (B) 1 (C) 0 (D) not defined
The highest order derivative present is $\frac{d^2y}{dx^2}$, which is of order 2.
$\boxed{\text{(A) 2}}$