Janagan Kandasamy Assignment Assignment 3 due 10/05/2016 at 11:59pm EDT Z ENRG 213 Fall 2016 1. (1 point) Use the \"mixed partials\" check to see if the following differential equation is exact. If it is exact find a function F(x, y) whose level curves are solutions to the differential equation 4. (1 point) Find a family of solutions to the differential equation (x2 xy)dx + xdy = 0 (4ex sin(y) + y)dx + (3x)dy = 0 (To enter the answer in the form below you may have to rearrange the equation so that the constant is by itself on one side of the equation.) Then the solution in implicit form is the set of points (x, y) where ? F(x, y) = = constant F(x, y) = Answer(s) submitted: Answer(s) submitted: (incorrect) (score 0.5) 5. (1 point) Solve the following differential equation: 2. (1 point) Use the \"mixed partials\" check to see if the following differential equation is exact. ydx + xdy + xdx = 0. 1 x2 y2 If it is exact find a function F(x, y) whose differential, dF(x, y) gives the differential equation. That is, level curves F(x, y) = C are solutions to the differential equation: = constant. help (formulas) Answer(s) submitted: dy 2x4 + 2y = dx 2x + y2 (incorrect) First rewrite as 6. (1 point) Solve the following differential equation: M(x, y) dx + N(x, y) dy = 0 where M(x, y) = and N(x, y) = 3x , dy = 2xex 3y + 6x2 . dx . = constant. help (formulas) If the equation is not exact, enter not exact, otherwise enter in F(x, y) as the solution of the differential equation here = C. Answer(s) submitted: (incorrect) Answer(s) submitted: 7. (1 point) A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)yn () dx (incorrect) 3. (1 point) Solve the following differential equation: Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y1n transforms the Bernoulli equation into the linear equation (y x5 )dx + (x + y5 )dy = 0. du + (1 n)P(x)u = (1 n)Q(x). dx = constant. help (formulas) Answer(s) submitted: Consider the initial value problem (incorrect) xy0 + y = 3xy2 , y(1) = 5. 1 (a) This differential equation can be written in the form () with P(x) = , Q(x) = , and n= . (b) The substitution u = ear equation du + u= . dx 9. (1 point) Solve the initial value problem y 0 = (x + y 2)2 with y(0) = 0. (1) To solve this, we should use the substitution u= help (formulas) help (formulas) u0 = Enter derivatives using prime notation (e.g., you would dy enter y 0 for dx ). will transform it into the lin- (c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u: . u(1) = (2) After the substitution from the previous part, we obtain the following linear differential equation in x, u, u 0 . help (equations) (d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c). u(x) = . (3) The solution to the original initial value problem is described by the following equation in x, y. help (equations) (e) Finally, solve for y. y(x) = Answer(s) submitted: . Answer(s) submitted: (incorrect) dy = xy + 9x + 2y + 18. If 10. (1 point) Find a solution to dx necessary, use k to denote an arbitrary constant. help (equations) Answer(s) submitted: (incorrect) (incorrect) 8. (1 point) A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)yn . dx 11. (1 point) Solve the initial value problem 2yy 0 + 6 = y + 6x with y(0) = 7. (1) To solve this, we should use the substitution u= help (formulas) With this substitution, help (formulas) y= y0 = help (formulas) Enter derivatives using prime notation (e.g., you would dy enter y 0 for dx ). 2 Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y1n transforms the Bernoulli equation into the linear equation du + (1 n)P(x)u = (1 n)Q(x). dx (2) After the substitution from the previous part, we obtain the following linear differential equation in x, u, u 0 . help (equations) Use an appropriate substitution to solve the equation 4 y4 y0 y = 2 , x x (3) The solution to the original initial value problem is described by the following equation in x, y. help (equations) and find the solution that satisfies y(1) = 1. y(x) = Answer(s) submitted: . Answer(s) submitted: (incorrect) 2 (incorrect) Solve the initial value problem y 0 = (t + y)2 1, y(2) = 5. 12. (1 point) Sometimes a change of variable can be used to convert a differential equation y 0 = f (t, y) into a separable equation. One common change of variable technique is as follows. (1) Consider a differential equation of the form y 0 = f (t + y + ), where , , and are constants. Use the change of variable z = t + y + to rewrite the differential equation as a separable equation of the form z 0 = g(z). (a) g(z) = (b) y(t) = Answer(s) submitted: (incorrect) c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 3 help (formulas) help (formulas)