Determine the order and degree (if defined) of the differential equations given in the following: 1. $\dfrac{{{d^4}y}}{{d{x^4}}} + \sin ({y^{'''}}) = 0$ 2. $y' + 5y = 0$ 3. ${\left( {\dfrac{{ds}}{{dt}}} \right)^4} + 3s\dfrac{{{d^2}s}}{{d{t^2}}} = 0$ 4. ${\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} - \cos \left( {\dfrac{{dy}}{{dx}}} \right) = 0$ 5. $\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right) = \cos 3x + \sin 3x$ 6. ${\left( {{y^{'''}}} \right)^2} + {\left( y \right)^3} + {\left( y \right)^4} + {y^5} = 0$ 7. $y + 2{y^{''}} + y = 0$ 8. $y' + y = {e^x}$ 9. ${y^{''}} + (y)^2 + \sin y = 0$ 10.${y^{''}} + 2y + \sin y = 0$
ANSWER
Verified
Hint: Let's make use of the definition of order and degree of a derivative and lets try to solve this problem. Order of a differential equation refers to the highest numbered derivative in the equation and degree refers to the power to which the highest numbered derivative is raised.
Complete step-by-step answer: Now let us solve the equations give 1. $\dfrac{{{d^4}y}}{{d{x^4}}} + \sin ({y^{'''}}) = 0$ Ans: In this case, the highest numbered derivative is 4 and it is raised to power of 1. So, Order=4 Degree=1
2. $y' + 5y = 0$ Ans: In this case, the highest order derivative is 1 and it is raised to the power of 1 So, Order=1 Degree=1
3. ${\left( {\dfrac{{ds}}{{dt}}} \right)^4} + 3s\dfrac{{{d^2}s}}{{d{t^2}}} = 0$ Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1 So, Order=2 Degree=1
4. ${\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} - \cos \left( {\dfrac{{dy}}{{dx}}} \right) = 0$ Ans: In this case,the highest order derivative is 2 and it is raised to the power of 2 So, Order=2 Degree=2
5. $\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right) = \cos 3x + \sin 3x$ Ans: In this case,the highest order derivative is 2 and it is raised to the power of 1 So, Order=2 Degree=1
6. ${\left( {{y^{'''}}} \right)^2} + {\left( y \right)^3} + {\left( y \right)^4} + {y^5} = 0$ Ans: In this case, the highest order derivative is 3 and it is raised to the power of 2 So, Order=3 Degree=2
7. $y + 2{y^{''}} + y = 0$ Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1 So, Order=2 Degree=1
8. $y' + y = {e^x}$ Ans: In this case, the highest order derivative is 1 and it is raised to the power of 1 So, Order=1 Degree=1
9. ${y^{''}} + (y)^2 + \sin y = 0$ Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1 So, Order=2 Degree=1
10. ${y^{''}} + 2y + \sin y = 0$ Ans: In this case, the highest order derivative is 2 and it is raised to the power of 1 So, Order=2 Degree=1
Note: In these types of questions it has to be noted that the order of the differential equation is the highest order derivative and not the highest power in the equation.