Differential Equations: Obtain the differential equation of the family of curves, parabolas with vertex and focus on axis. а. (х - у)у" — у'з — у? — у' — 1%3о b. (x – y)² (y'² + 1) = (x + yy')² c. xy" – y' = 0 d. хуу" + ху? —- уу' %— 0
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- Van der Waals Equation In Exercise 18 at the end of Section 2.3, we discussed the ideal gas law, which shows the relationship among volume V, pressure p, and temperature T for a fixed amount 1 mole of a gas. But chemists believe that in many situations, the van der Waals equation gives more accurate results. If we measure temperature T in kelvins, volume V in liters, and pressure p in atmosphere 1 atm is the pressure exerted by the atmosphere at sea level, then the relationship for carbon dioxide is given by p=0.082TV0.0433.592V2atm What volume does this equation predict for 1 mole of carbon dioxide at 500 kelvins and 100 atm?Suggestion: Consider volumes ranging from 0.1 to 1 liter.a. Obtain the differential equation having the solution as the equation representing all parabolas with vertex and focus on the x-axis opening to the right. b. Find the differential equation having the solution as the equation representing all lines with slope and y-intercepts equal.Application of Differential Equation(mixing problem): A tank initially contains 100 L of a well-mixed water-and-salt solution, with salt concentration of 20 kg in the solution. If the solution is exiting from the tank at a constant rate of 2 L per minute, what is the concentration of the salt (kilograms) in the solution at the point when the tank now only contains 50 L of the solution?
- It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation -- (what is the highest number of derivatives involed) and whether or not the equation is linear. Linearity is important because the structure of the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear for 2 3 and 4 it is also based on the drop down menu (in the picture)Differential EquationsFind the Particular Solution of (16x+5y)dx + (3x+y)dy = 0. If the curve to pass through the point (1, -3)Even before you learn techniques for solving differential equations, you may be able to analyze equations qualitatively . As an example, look at the nonlinear equationdydt=(y−4)2(y+5)You are going to analyze the solutions, y, of this equation without actually finding them. You will be asked to sketch three solutions of the differential equation on the graph below based on qualitative information from the differential equation. In what follows, picture the t-axis running horizontally and the y axis running vertically. There is no scale on the t axis but imagine it is large enough to display the behavior of the solutions as t approaches ±∞. a) For what values of y is the graph of y as a function of t increasing? Use interval notation for your answer. b) For what values of y is the graph of y concave up? For what values of y is it concave down? (Help with interval notation.)What information do you need to answer a question about concavity? Remember that y is an implicit function of t.…