Report on differential equation
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This document discusses geometrical applications of differential equations and orthogonal trajectories. It provides examples of using differential equations to find the equation of a curve given properties of its tangent lines. It also explains that the differential equation describing the family of curves orthogonal to a given family can be found by taking the reciprocal of the differential equation for the original family. Several problems are presented as examples, such as finding the orthogonal trajectories to families of circles, exponential curves, and rectangular hyperbolas.
Report on differential equation
- 1. Geometrical Applications of Differential Equations & Orthogonal Trajectories
- 2. P T M G x y y = f(x) (x0, y0) tangent norma l subtange nt subnorm al
- 3. The slope of a tangent to the curve at any point is the reciprocal of the y-coordinate of that point (y ≠ 0) and the curve passes through (-1, 2). Find the equation of the curve.
- 4. Let P (x, y) be the point given on the curve. As per given condition, slope of the tangent = 1 𝑦 𝑑𝑦 𝑑𝑥 = 1 𝑦 𝑦𝑑𝑦 = 𝑑𝑥 𝑦𝑑𝑦 = 𝑑𝑥 𝑦2 2 = 𝑥 + 𝑐′ 𝑦2 = 2𝑥 + 2𝑐′ 𝑦2 = 2𝑥 + 𝑐 where 𝑐 = 2𝑐′ Solving for c using (-1, 2), c = 6. Therefore 𝑦2 = 2𝑥 + 6 is the required equation of the curve.
- 5. 1. The tangent line to a curve at any point (x, y) on it has its intercept on the x-axis always equal to x 2 . If the curve passes through (1, 2), find its equation. 2. Find the curve through point (1, 1) in the xy- plane having at each of its points the slope − y x . 3. The slope at any point of a curve is 2x + 3y. If the curve passes through the origin, determine its equation.
- 6. If we are given a family of curves we may think of another family of curves such that each member of this family cuts each member of another family at right angles. x y A B L M N O P
- 7. Orthogonal trajectories of the family of circles having center at the origin are the members of the family of straight lines.
- 8. To find orthogonal trajectories of a given family of curves, we first find the differential equation dy dx = f(x, y) which describes the family. The differential equation of the second and orthogonal family is then dy dx = − 1 f(x,y) .
- 9. 1. Find an equation of the family orthogonal to the family y = cx. 2. Find the family orthogonal to the family 𝑦 = 𝑐𝑒−𝑥 of exponential curves. Determine the member of each family passing through (0, 4).
- 10. 1. Find the orthogonal trajectories 𝑥2 + 𝑦2 = 𝑐𝑥. 2. Find the orthogonal trajectories of the family 𝑦 = 𝑥 + 𝑐𝑒−𝑥 and determine that particular member of each family that passes through (0, 3). 3. Find the orthogonal trajectories of the family of rectangular hyperbolas
- 11. Ahsan, Z. (2006). Differential equations and their applications. New Delhi: Prentice- Hall of India Private Limited. Thank you for listening. Raymund T. de la Cruz MAEd - Mathematics
- 12. y – y0 = m(x – x0) y – y0 = 𝑑𝑦 𝑑𝑥 (x – x0)
- 13. y – y0 = − 1 𝑚 (x – x0) y – y0 = − 𝑑𝑥 𝑑𝑦 (x – x0)
- 14. 𝑇𝑀 = 𝑦0 𝑑𝑦 𝑑𝑥 𝑇𝑀 = 𝑦0 𝑑𝑥 𝑑𝑦
- 15. 𝑀𝐺 = 𝑦0 𝑑𝑦 𝑑𝑥