Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 10 Ordinary Differential Equations Ex 10.1 Textbook Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.1

Question 1.

For each of the following equations, determine its order, degree (if exists)

(i) \(\frac { dy }{ dx }\) + xy = cot x

(ii) (\(\frac { d^3y }{ dx^3 }\))^{2/3} – 3 \(\frac { d^2y }{ dx^2 }\) + 5\(\frac { dy }{ dx }\) + 4 = 0

(iii) (\(\frac { d^2y }{ dx^2 }\))^{2} + (\(\frac { dy }{ dx }\))² = x sin (\(\frac { d^2y }{ dx^2 }\))

(iv) \(\sqrt{\frac { dy }{ dx }}\) – 4 \(\frac { dy }{ dx }\) – 7x = 0

(v) y(\(\frac { dy }{ dx }\)) = \(\frac { x }{ (\frac { dy }{ dx })+(\frac { dy }{ dx })^3 }\)

(vi) x²\(\frac { d^2y }{ dx^2 }\) + [1 + (\(\frac { dy }{ dx }\))²]^{1/2} = 0

(vii) (\(\frac { d^2y }{ dx^2 }\))³ = \(\sqrt{1+(\frac { dy }{ dx })}\)

(viii) \(\frac { d^2y }{ dx^2 }\) = xy + cos (\(\frac { dy }{ dx }\))

(ix) \(\frac { d^2y }{ dx^2 }\) + 5 \(\frac { dy }{ dx }\) + ∫ ydx = x³

(x) x = e^{xy(\(\frac { dy }{ dx }\))}

Solution:

(i) \(\frac { dy }{ dx }\) + xy = cot x

In the given equation, the highest order derivative is \(\frac { dy }{ dx }\) only its power is 1

∴ Its order = 1 & degree = 1

(ii) (\(\frac { d^3y }{ dx^3 }\))^{2/3} – 3 \(\frac { d^2y }{ dx^2 }\) + 5\(\frac { dy }{ dx }\) + 4 = 0

Taking power 3 on both sides, we get

(\(\frac { d^3y }{ dx^3 }\))^{2} = (3 \(\frac { d^2y }{ dx^2 }\) – 5\(\frac { dy }{ dx }\) – 4)³

In the equation (1), the highest order derivative is \(\frac { d^3y }{ dx^3 }\) and its power is 2.

∴ Its order = 3 & degree = 2

(iii) (\(\frac { d^2y }{ dx^2 }\))^{2} + (\(\frac { dy }{ dx }\))² = x sin (\(\frac { d^2y }{ dx^2 }\))

In the equation, the highest order derivative is \(\frac { d^2y }{ dx^2 }\) and its order is 2.

It has a term sin (\(\frac { d^2y }{ dx^2 }\)), so its degree is not defined or degree does not exist.

on squaring both sides,

\(\frac { dy }{ dx }\) = 16 (\(\frac { dy }{ dx }\))² + 49 x² + 56x \(\frac { dy }{ dx }\)

clearly, it is a differential equation of order = 1 & degree = 2.

In this equation, the highest order derivative is \(\frac { dy }{ dx }\) & its power is 4.

∴ Its order = 1 & degree = 4

In this equation, the highest order derivative is \(\frac { d^2y }{ dx^2 }\) & its power is 2.

∴ Its order = 2 & degree = 2

In this equation, the highest order derivative is \(\frac { d^2y }{ dx^2 }\) & its power is 6.

∴ Its order = 2 & degree = 6

(viii) \(\frac { d^2y }{ dx^2 }\) = xy + cos (\(\frac { dy }{ dx }\))

In this equation, the highest order derivative is \(\frac { d^2y }{ dx^2 }\) & its power is 2.

It has a term cos(\(\frac { dy }{ dx }\)), so its degree is not defined or degree does not exist.

(ix) \(\frac { d^2y }{ dx^2 }\) + 5 \(\frac { dy }{ dx }\) + ∫ ydx = x³

differentiating with respect to x, we get

\(\frac { d^3y }{ dx^3 }\) + 5 \(\frac { d^2y }{ dx^2 }\) + y = 3x²

In this equation the highest order derivative is \(\frac { d^3y }{ dx^3 }\) & its power is 1

∴ Its order = 3 & degree = 1

(x) x = e^{xy(\(\frac { dy }{ dx }\))}

In this equation the highest order derivative is \(\frac { dy }{ dx }\) & its order is 1

It has the term e^{xy(\(\frac { dy }{ dx }\))}

So its degree is not defined or degree does not exist.