Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.6

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Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.6

Solve the following differential equations:

Question 1.

Solution:

This is a Homogeneous differential equation

Seperating the variables

Question 2.
(x³ + y³) dy – x²y dx = 0
Solution:
(x³ + y³) dy – x²y dx = 0
(x³ + y³) dy = x³y dx

This is a Homogeneous differential equation.

Separating the variables

Question 3.

Solution:

This is a Homogeneous differential equation

Seperating the variables

Question 4.
2xydx + (x² + 2y²) dy = 0
Solution:
2xy dx + (x² + 2y²) dy = 0
(x² + 2y²) dy = – 2xy dx

This is a Homogeneous differential equation


Separating the variables

Question 5.
(y² – 2xy)dx = (x² – 2xy)dy
Solution:
(y² – 2xy) dx = (x² – 2xy) dy

This is a Homogeneous differential equation


Seperating the variables

Question 6.

Solution:

Seperating the variables

Question 7.

Solution:

This is a Homogeneous differential equation

Separating the variables

Question 8.
(x² + y²) dy = xy dx. It is given that y(1) = 1 and y(x₀) = e. Find the value of x₀.
Solution:

This is a Homogeneous differential equation

Seperating the variables

Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.6 Additional problems

Question 1.
Solve: (2\(\sqrt{x y}\))dy + y dx = 0
Solution:

Question 2.
Solve: (x³ + 3xy²)dx + (y³ + 3x²y)dy = 0
Solution:


Integrating, we have

Question 3.

Solution:

Which is in terms of v alone.
⇒ the given problem comes under homogeneous type.

Question 4.
Solve: (x² + y²)dy = xy dx
Solution:

Which is a function in v alone.
⇒ the given problem comes under homogeneous type.

Question 5.
Find the equation of the curve passing through (1, 0) and which has slope \(1+\frac{y}{x}\) at (x, y).
Solution:


Given the curve passes through (1, 0) ⇒ at x = 1, y = 0
0 = log 1 +c ⇒ c = 0