Determine the order and degree if defined of the following differential equations-i d s d t 4-3 sd 2 sdt 2-0ii y-2 y-y-0iii y-2-y-3-y-4-y5-0iv y-2 y-y-0v y-y-2-2 y-0vi y-2 y-sin y-0vii y-y2-ey-0

I #160; #160;dsdt4+3sd2sdt2=0 The highest order derivative in the given equation is d2sdt2 and its power is 1. Therefore, the given differential equation is ...

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Standard XII

Mathematics

Derivative Operator dy/dx

Determine the...

Question

Determine the order and degree (if defined) of the following differential equations:
(i) ${(\frac{d s}{d t})}^{4} + 3 s \frac{d^{2} s}{d t^{2}} = 0$

(ii) y"' + 2y" + y' = 0
(iii) (y"')² + (y")³ + (y')⁴ + y⁵ = 0
(iv) y"' + 2y" + y' = 0
(v) y" + (y')² + 2y = 0
(vi) y" + 2y' + sin y = 0
(vii) y"' + y² + e^y^' = 0

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Solution

$(i) {(\frac{d s}{d t})}^{4} + 3 s \frac{d^{2} s}{d t^{2}} = 0$
The highest order derivative in the given equation is $\frac{d^{2} s}{d t^{2}}$ and its power is 1.
Therefore, the given differential equation is of second order and first degree.
i.e., Order = 2 and degree = 1

(ii) y"' + 2y" + y' = 0
The highest order derivative in the given equation is y''' and its power is 1.
Therefore, the given differential equation is of third order and first degree.
i.e., Order = 3 and degree = 1

(iii) (y"')² + (y")³ + (y')⁴ + y⁵ = 0
The highest order derivative in the given equation is y''' and its power is 2.
Therefore, the given differential equation is of third order and second degree.
i.e., Order = 3 and degree = 2

(iv) y"' + 2y" + y' = 0
The highest order derivative in the given equation is y''' and its power is 1.
Therefore, the given differential equation is of third order and first degree.
i.e., Order = 3 and degree = 1

(v) y" + (y')² + 2y = 0
The highest order derivative in the given equation is y'' and its power is 1.
Therefore, the given differential equation is of second order and first degree.
i.e., Order = 2 and degree = 1

(vi) y" + 2y' + sin y = 0
The highest order derivative in the given equation is y'' and its power is 1.
Therefore, the given differential equation is of second order and first degree.
i.e., Order = 2 and degree = 1

(vii) y"' + y² + e^y^' = 0
The highest order derivative in the given equation is y''' and its power is 1.
Therefore, the given differential equation is of third order. This equation cannot be expressed as a polynomial of derivative.
Thus, the degree is not defined.
i.e., Order = 3 and degree is not defined.

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Determine the order and degree (if defined) of the following differential equations: (i) dsdt4+3sd2sdt2=0 (ii) y"' + 2y" + y' = 0 (iii) (y"')2 + (y")3 + (y')4 + y5 = 0 (iv) y"' + 2y" + y' = 0 (v) y" + (y')2 + 2y = 0 (vi) y" + 2y' + sin y = 0 (vii) y"' + y2 + ey' = 0