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. ’’’''
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.