Q 1: Determine the order and degree (if defined) of the following differential equation: \frac{d^{4}y}{dx^{4}} + \sin(y''') = 0
Solution:
Step 1: Identify the Highest Order Derivative
Looking at the equation, we have a fourth-order derivative term: \frac{d^{4}y}{dx^{4}} .
We also have a lower order derivative inside the sine function.
The highest order derivative is 4.
Therefore, the Order is 4.
Step 2: Check the Polynomial Condition for Degree
To find the Degree, the differential equation must be a polynomial equation in its derivatives (like y', y'', etc.).
We examine the term: \sin(y''') .
Because the derivative is inside a trigonometric function (sine), this equation cannot be expressed as a polynomial in derivatives.
Therefore, the Degree is Not Defined.
Final Answer: \boxed{ \text{Order: } 4, \text{ Degree: Not Defined} }
Q 2: Determine the order and degree (if defined) of the following differential equation: y' + 5y = 0
Solution:
Step 1: Identify the Highest Order Derivative
The equation involves y' , which is the first derivative \frac{dy}{dx} .
There are no higher derivatives.
Therefore, the Order is 1.
Step 2: Check the Polynomial Condition for Degree
To find the Degree, the equation must be a polynomial in its derivatives.
Here, the equation is y' + 5y = 0 .
It is a simple polynomial equation in terms of the derivative y' .
The highest power (exponent) of the highest order derivative ( y' ) is 1.
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 1, \text{ Degree: } 1 }
Q 3: Determine the order and degree (if defined) of the following differential equation: (\frac{ds}{dt})^{4} + 3s \frac{d^{2}s}{dt^{2}} = 0
Solution:
Step 1: Identify the Highest Order Derivative
The equation contains two derivative terms: \frac{ds}{dt} (first order) and \frac{d^{2}s}{dt^{2}} (second order).
Comparing the two, the highest order derivative is \frac{d^{2}s}{dt^{2}} .
Therefore, the Order is 2.
Step 2: Check the Polynomial Condition for Degree
We verify if the equation is a polynomial in its derivatives.
The equation contains powers of derivatives ( (\frac{ds}{dt})^4 and \frac{d^{2}s}{dt^{2}} ), but no derivatives are inside special functions like sine, log, or exponential.
Thus, the degree is defined.
We look at the power of the highest order derivative.
The highest order derivative is \frac{d^{2}s}{dt^{2}} , and its power is 1. (Note: We ignore the power 4 on the lower order derivative).
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 2, \text{ Degree: } 1 }
Q 4: Determine the order and degree (if defined) of the following differential equation: (\frac{d^{2}y}{dx^{2}})^{2} + \cos(\frac{dy}{dx}) = 0
Solution:
Step 1: Identify the Highest Order Derivative
The derivatives present are \frac{d^{2}y}{dx^{2}} and \frac{dy}{dx} .
The highest order derivative is \frac{d^{2}y}{dx^{2}} , which is of order 2.
Therefore, the Order is 2.
Step 2: Check the Polynomial Condition for Degree
We check if the equation can be expressed as a polynomial in derivatives.
We observe the term \cos(\frac{dy}{dx}) .
Since a derivative, \frac{dy}{dx} , appears inside a trigonometric function (cosine), the equation is not a polynomial in its derivatives.
Therefore, the Degree is Not Defined.
Final Answer: \boxed{ \text{Order: } 2, \text{ Degree: Not Defined } }
Q 5: Determine the order and degree (if defined) of the following differential equation: \frac{d^{2}y}{dx^{2}} = \cos 3x + \sin 3x
Solution:
Step 1: Identify the Highest Order Derivative
The only derivative present in the equation is \frac{d^{2}y}{dx^{2}} .
This is a second-order derivative.
Therefore, the Order is 2.
Step 2: Check the Polynomial Condition for Degree
We check for polynomial structure in terms of derivatives.
The trigonometric terms \cos 3x and \sin 3x involve only the independent variable x , not the derivatives of y .
Therefore, they do not affect the definition of the degree.
The equation is a polynomial in the derivative \frac{d^{2}y}{dx^{2}} .
The power of the highest order derivative ( \frac{d^{2}y}{dx^{2}} ) is 1.
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 2, \text{ Degree: } 1 }
Q 6: Determine the order and degree (if defined) of the following differential equation: (y''')^{2} + (y'')^{3} + (y')^{4} + y^{5} = 0
Solution:
Step 1: Identify the Highest Order Derivative
We examine all the derivative terms in the equation: y''' (third derivative), y'' (second derivative), and y' (first derivative).
The highest order derivative is y''' .
Therefore, the Order is 3.
Step 2: Check the Polynomial Condition for Degree
The equation is a polynomial equation in its derivatives (y''', y'', y') because no derivative appears inside functions like sine, log, or exponential.
To find the degree, we look at the highest power (exponent) of the highest order derivative.
The highest order derivative is y''' , and its power is 2.
Therefore, the Degree is 2.
Final Answer: \boxed{ \text{Order: } 3, \text{Degree: } 2 }
Q 7: Determine the order and degree (if defined) of the following differential equation: y''' + 2y'' + y' = 0
Solution:
Step 1: Identify the Highest Order Derivative
The equation contains the derivatives y''' , y'' , and y' .
The highest order derivative is y''' (the third derivative).
Therefore, the Order is 3.
Step 2: Check the Polynomial Condition for Degree
This is a polynomial equation in derivatives.
We look at the power of the highest order derivative y''' .
The term is simply y''' , which implies a power of 1.
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 3, \text{Degree: } 1 }
Q 8: Determine the order and degree (if defined) of the following differential equation: y' + y = e^{x}
Solution:
Step 1: Identify the Highest Order Derivative
The only derivative present is y' (first derivative).
Therefore, the Order is 1.
Step 2: Check the Polynomial Condition for Degree
The equation is a polynomial in terms of the derivative y' .
Note that e^x is a function of the independent variable x and does not affect the polynomial condition regarding derivatives.
The highest power of the highest derivative ( y' ) is 1.
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 1, \text{Degree: } 1 }
Q 9: Determine the order and degree (if defined) of the following differential equation: y'' + (y')^{2} + 2y = 0
Solution:
Step 1: Identify the Highest Order Derivative
The derivatives involved are y'' and y' .
The highest order derivative is y'' (second order).
Therefore, the Order is 2.
Step 2: Check the Polynomial Condition for Degree
The equation involves (y')^2 , but this is allowed in a polynomial equation.
We strictly look at the power of the highest order derivative, which is y'' .
The power of y'' is 1.
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 2, \text{Degree: } 1 }
Q 10: Determine the order and degree (if defined) of the following differential equation: y'' + 2y' + \sin y = 0
Solution:
Step 1: Identify the Highest Order Derivative
The highest order derivative in the equation is y'' .
Therefore, the Order is 2.
Step 2: Check the Polynomial Condition for Degree
We check if the equation is a polynomial in its derivatives.
We see the term \sin y .
Since the sine function is applied to the dependent variable y and not to a derivative (like \sin(y') ), the polynomial condition is satisfied.
The highest power of the highest derivative ( y'' ) is 1.
Therefore, the Degree is 1.
Final Answer: \boxed{ \text{Order: } 2, \text{Degree: } 1 }
Q 11: The degree of the differential equation (\frac{d^{2}y}{dx^{2}})^{3} + (\frac{dy}{dx})^{2} + \sin(\frac{dy}{dx}) + 1 = 0 is
(A) 3
(B) 2
(C) 1
(D) not defined
Solution:
Step 1: Check the Polynomial Condition for Degree
To find the degree, the differential equation must be a polynomial equation in its derivatives (like y', y'', etc.).
We observe the term \sin(\frac{dy}{dx}) in the given equation.
Because the first derivative \frac{dy}{dx} is trapped inside a trigonometric function (sine), this equation cannot be expressed as a polynomial in its derivatives.
Step 2: Conclusion Since the equation fails the polynomial test, the degree is not defined.
Final Answer: The correct option is (D).
Q 12: The order of the differential equation 2x^{2}\frac{d^{2}y}{dx^{2}} - 3\frac{dy}{dx} + y = 0 is
(A) 2
(B) 1
(C) 0
(D) not defined
Solution:
Step 1: Identify the Highest Order Derivative
We look at all the derivative terms present in the equation:
\frac{dy}{dx} is a first-order derivative.
\frac{d^{2}y}{dx^{2}} is a second-order derivative.
Step 2: Determine the Order
The order of a differential equation is determined by the highest order derivative involved.
Here, the highest order derivative is \frac{d^{2}y}{dx^{2}} , which is of order 2.
Final Answer: The correct option is (A).