A First Course in Ordinary Differential Equations by Rudolph E. Langer

By Rudolph E. Langer

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4, the path of the pursuing plane A is a curve expressible by an equation either i n the polar coordinates (r, 6) or i n the rectangular coordinates (x — x*, y — y*). With reference to the ground, A has a velocity whose eastward component dx/dt is made up of the air-speed component V cos (x — 0) and the wind component Vx. T h u s dx/dt = — Kcos 0 + Vx. Applications 50 Similarly, A^s northward velocity component is dy/dt = - F sin + Vy. For the plane 0 we have dx*/dt = a, and dy*/dt = b. 12) [VsmB-Vy-\-b]d{x-x*\ - \Vcos6 ~ ^ a}d\y -y*\ = 0.

Example 9. T o find the equation of the curve that goes through the point (2,1), such that the area under any of its arcs and above the x-axis is equal to the length of the respective arc. F r o m the calculus we know that, for a curve above the x-axis, which extends to the right from a point (XQ, >O)» the area under an arc is given by the integral / y dx. T h e length of the corresponding arc is T h e curve sought is, therefore, one for which B y differentiation this becomes y y' = + y/y^ — 1.

41. A n 84b. body is made of metal whose specific heat is ^ T J . While at a temperature of 308 degrees it is plunged into 11 lb, of water at a temperature of 53 degrees. Find the temperature to which the body eventually cools. 44 Applications 42. A 10-lb. body whose specific heat is -g- is at 0 degrees when t ^ 0. It is then plunged into 100 lb- of water at 80 degrees. 6 applies with k — yf* what is the temperature of the body at time t? 43. A 5-lb. body whose specific heat is TnJ" is plunged at / 0 into 50 lb.

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