Differential Equations - Orthogonal Trajectories

• An orthogonal trajectory is a curve that intersects another curve at a right angle.
• In differential equations, we will study how to find the orthogonal trajectories of a given curve.
• Orthogonal trajectories are obtained by solving a different differential equation.

Orthogonal Trajectories - Definition

• Let’s consider a family of curves given by an equation f(x, y, C) = 0, where C is a constant.
• The orthogonal trajectories of this family of curves are the curves that intersect each curve of the family at a right angle.
• We can find them by solving a different differential equation.

Orthogonal Trajectories - Finding

• To find the orthogonal trajectory, differentiate the given equation with respect to x, with y as the dependent variable.
• Let dy/dx represent the slope of the curve.
• The slope of the orthogonal trajectory will be the negative reciprocal of dy/dx, i.e., -dx/dy.
• Solve the differential equation dy/dx = -dx/dy to obtain the orthogonal trajectory.

Orthogonal Trajectories - Example

Given equation: x^2 + y^2 = c, where c is a constant.

• Step 1: Differentiate the equation with respect to x.
• 2x + 2yy’ = 0, where y’ represents dy/dx.
• Simplifying, we get y’ = -x/y.

Orthogonal Trajectories - Example (contd.)

To find the orthogonal trajectory:

• Step 2: Take the negative reciprocal of y'.
• -dx/dy = -y/x
• Step 3: Solve the differential equation -dx/dy = -y/x
• Rearranging, we get dy/dx = y/x.

Orthogonal Trajectories - Example (contd.)

• Step 4: Solve the differential equation dy/dx = y/x using separation of variables.
• Separate variables: dy/y = dx/x
• Integrating both sides, we get ln|y| = ln|x| + ln|c|
• ln|y| = ln|cx|
• Taking exponential on both sides, we get |y| = cx

Orthogonal Trajectories - Example (contd.)

• Step 5: Remove the absolute value sign by considering positive and negative values.
• y = cx, where c can be a positive or negative constant.
• Thus, the orthogonal trajectories of the given family of curves are straight lines passing through the origin.

Orthogonal Trajectories - Summary

• Orthogonal trajectories are curves that intersect another curve at a right angle.
• To find orthogonal trajectories, we differentiate the given equation with respect to x, with y as the dependent variable.
• We solve the obtained differential equation to find the orthogonal trajectories.
• In the given example, the orthogonal trajectories were straight lines passing through the origin.

Orthogonal Trajectories - Example

Given equation: y^2 = 4ax, where a is a constant.

• Step 1: Differentiate the equation with respect to x.
• 2yy’ = 4a
• Step 2: Solve for y'.
• y’ = 2a/y
• Step 3: Take the negative reciprocal of y'.
• -dx/dy = -y/(2a)
• Step 4: Rearrange the equation.
• dy/dx = -2a/y
• Step 5: Solve the differential equation dy/dx = -2a/y.
• Separate variables: ydy = -2adx
• Integrating both sides: (1/2) y^2 = -2ax + c
• Simplifying: y^2 = -4ax + 2c

Orthogonal Trajectories - Example (contd.)

• Step 6: Removing the constant.
• y^2 + 4ax = C, where C is a constant.
• Step 7: Interpreting the result.
• The orthogonal trajectories of the given family of curves are parabolas in the x-y plane.

Orthogonal Trajectories - Example

Given equation: y^2 = x^3

• Step 1: Differentiate the equation with respect to x.
• 2yy’ = 3x^2
• Step 2: Solve for y'.
• y’ = (3x^2) / (2y)
• Step 3: Take the negative reciprocal of y'.
• -dx/dy = - (2y) / (3x^2)
• Step 4: Rearrange the equation.
• dy/dx = (3x^2) / (2y)
• Step 5: Solve the differential equation dy/dx = (3x^2) / (2y).
• Separate variables: ydy = (3x^2)dx / 2
• Integrating both sides: (1/2) y^2 = (3/2) x^3 + c
• Simplifying: y^2 = 3x^3 + 2c

Orthogonal Trajectories - Example (contd.)

• Step 6: Removing the constant.
• y^2 = 3x^3 + C, where C is a constant.
• Step 7: Interpreting the result.
• The orthogonal trajectories of the given family of curves are cubic curves in the x-y plane.

Orthogonal Trajectories - Example

Given equation: y = ce^x

• Step 1: Differentiate the equation with respect to x.
• dy/dx = ce^x
• Step 2: Take the negative reciprocal of dy/dx.
• -dx/dy = e^-x / c
• Step 3: Rearrange the equation.
• dy/dx = -c / e^-x
• Step 4: Solve the differential equation dy/dx = -c / e^-x
• Separate variables: dy = -c / e^-x dx
• Integrating both sides: y = ce^-x + k, where k is a constant

Orthogonal Trajectories - Example (contd.)

• Step 5: Interpreting the result.
• The orthogonal trajectories of the given family of curves are exponential curves in the x-y plane.
• The general equation for the orthogonal trajectories is y = ce^-x + k, where c and k are constants.

Orthogonal Trajectories - Example

Given equation: y^2 = cx, where c is a constant.

• Step 1: Differentiate the equation with respect to x.
• 2yy’ = c
• Step 2: Solve for y'.
• y’ = c / (2y)
• Step 3: Take the negative reciprocal of y'.
• -dx/dy = - (2y) / c
• Step 4: Rearrange the equation.
• dy/dx = (2y) / c
• Step 5: Solve the differential equation dy/dx = (2y) / c.
• Separate variables: ydy = (2y)dx / c
• Integrating both sides: (1/2) y^2 = (2/3) x^3 + k
• Simplifying: y^2 = (4/3) x^3 + 2k

Orthogonal Trajectories - Example (contd.)

• Step 6: Removing the constant.
• y^2 = (4/3) x^3 + C, where C is a constant.
• Step 7: Interpreting the result.
• The orthogonal trajectories of the given family of curves are curves in the x-y plane.
• The general equation for the orthogonal trajectories is y^2 = (4/3) x^3 + C, where C is a constant.

‘‘‘‘Mark Down

Differential Equations - Orthogonal Trajectories

• An orthogonal trajectory is a curve that intersects another curve at a right angle.
• In differential equations, we will study how to find the orthogonal trajectories of a given curve.
• Orthogonal trajectories are obtained by solving a different differential equation.

Orthogonal Trajectories - Definition

• Let’s consider a family of curves given by an equation f(x, y, C) = 0, where C is a constant.
• The orthogonal trajectories of this family of curves are the curves that intersect each curve of the family at a right angle.
• We can find them by solving a different differential equation.

Orthogonal Trajectories - Finding

• To find the orthogonal trajectory, differentiate the given equation with respect to x, with y as the dependent variable.
• Let dy/dx represent the slope of the curve.
• The slope of the orthogonal trajectory will be the negative reciprocal of dy/dx, i.e., -dx/dy.
• Solve the differential equation dy/dx = -dx/dy to obtain the orthogonal trajectory.

Orthogonal Trajectories - Example

Given equation: x^2 + y^2 = c, where c is a constant.

• Step 1: Differentiate the equation with respect to x.
• 2x + 2yy’ = 0, where y’ represents dy/dx.
• Simplifying, we get y’ = -x/y.

Orthogonal Trajectories - Example (contd.)

To find the orthogonal trajectory:

• Step 2: Take the negative reciprocal of y’.
• -dx/dy = -y/x
• Step 3: Solve the differential equation -dx/dy = -y/x
• Rearranging, we get dy/dx = y/x.

Orthogonal Trajectories - Example (contd.)

• Step 4: Solve the differential equation dy/dx = y/x using separation of variables.
• Separate variables: dy/y = dx/x
• Integrating both sides, we get ln|y| = ln|x| + ln|c|
• ln|y| = ln|cx|
• Taking exponential on both sides, we get |y| = cx

Orthogonal Trajectories - Example (contd.)

• Step 5: Remove the absolute value sign by considering positive and negative values.
• y = cx, where c can be a positive or negative constant.
• Thus, the orthogonal trajectories of the given family of curves are straight lines passing through the origin.

Orthogonal Trajectories - Summary

• Orthogonal trajectories are curves that intersect another curve at a right angle.
• To find orthogonal trajectories, we differentiate the given equation with respect to x, with y as the dependent variable.
• We solve the obtained differential equation to find the orthogonal trajectories.
• In the given example, the orthogonal trajectories were straight lines passing through the origin.

Orthogonal Trajectories - Example

Given equation: y^2 = 4ax, where a is a constant.

• Step 1: Differentiate the equation with respect to x.
• 2yy’ = 4a
• Step 2: Solve for y’.
• y’ = 2a/y
• Step 3: Take the negative reciprocal of y’.
• -dx/dy = -y/x
• Step 4: Rearrange the equation.
• dy/dx = y/x
• Step 5: Solve the differential equation dy/dx = y/x.
• Separate variables: ydy = xdx
• Integrating both sides: (1/2) y^2 = (1/2) x^2 + c
• Simplifying: y^2 = x^2 + c

Orthogonal Trajectories - Example (contd.)

• Step 6: Removing the constant.
• y^2 - x^2 = C, where C is a constant.
• Step 7: Interpreting the result.
• The orthogonal trajectories of the given family of curves are hyperbolas in the x-y plane.

Orthogonal Trajectories - Example

Given equation: y^2 = x^3

• Step 1: Differentiate the equation with respect to x.
• 2yy’ = 3x^2
• Step 2: Solve for y’.
• y’ = 3x^2 / (2y)
• Step 3: Take the negative reciprocal of y'.
• -dx/dy = -2y / (3x^2)
• Step 4: Rearrange the equation.
• dy/dx = 2y / (3x^2)
• Step 5: Solve the differential equation dy/dx = 2y / (3x^2).
• Separate variables: ydy = 2dx / (3x^2)
• Integrating both sides: (1/2) y^2 = -2 / (3x) + c
• Simplifying: y^2 = -4 / (3x) + 2c

Orthogonal Trajectories - Example (contd.)

• Step 6: Removing the constant.
• y^2 = -4 / (3x) + C, where C is a constant.
• Step 7: Interpreting the result.
• The orthogonal trajectories of the given family of curves are curves in the x-y plane.

Orthogonal Trajectories - Example

Given equation: y = ce^x

• Step 1: Differentiate the equation with respect to x.
• dy/dx = ce^x
• Step 2: Take the negative reciprocal of dy/dx.
• -dx/dy = e^-x / c
• Step 3: Rearrange the equation.
• dy/dx = -c / e^-x
• Step 4: Solve the differential equation dy/dx = -c / e^-x
• Separate variables: dy = -c / e^-x dx
• Integrating both sides: y = ce^-x + k, where k is a constant

Orthogonal Trajectories - Example (contd.)

• Step 5: Interpreting the result.
• The orthogonal trajectories of the given family of curves are exponential curves in the x-y plane.
• The general equation for the orthogonal trajectories is y = ce^-x + k, where c and k are constants.

Orthogonal Trajectories - Example

Given equation: y = cx, where c is a constant.

• Step 1: Differentiate the equation with respect to x.
• dy/dx = c
• Step 2: Take the negative reciprocal of dy/dx.
• -dx/dy = 1/c
• Step 3: Rearrange the equation.
• dy/dx = -1/c
• Step 4: Solve the differential equation dy/dx = -1/c
• Separate variables: dy = -1/c dx
• Integrating both sides: y = -x/c + k, where k is a constant

Orthogonal Trajectories - Example (contd.)

• Step 5: Interpreting the result.
• The orthogonal trajectories of the given family of curves are lines in the x-y plane.
• The general equation for the orthogonal trajectories is y = -x/c + k, where c and k are constants. ’’’''