Question

Consider the differential equation below, in which a,b,c,k are real constants and c² +k² >0 : (*) dxdy​ = cx+kyax+by​ .

(a) Find 4 specific numbers a,b,c,k for which the function y=x² −3x provides a solution for equation (∗) on the interval (0,+[infinity]). Briefly explain your method.
(b) Is there only one correct answer to part (a)? If so, explain why; if not, describe the set of all choices for (a,b,c,k) that should be accepted.
(c) Consider the ODE(∗) with the constants found in part (a). Find a solution satisfying y(2)=− 2549.

Answer 1

To find specific numbers a, b, c, k for which the function y=x²-3x provides a solution for the differential equation, we substitute y=x²-3x and dx=1 into the equation and solve for dy. The set of choices for (a, b, c, k) that should be accepted is any set of numbers that satisfy the equation. To find a solution satisfying the given condition in part (c), we substitute the values of a, b, c, and k found in part (a) into the differential equation, and solve for y(2)=-2549.

Step-by-step explanation:

To find specific numbers a, b, c, k for which the function y=x²−3x provides a solution for the differential equation, we substitute y=x²−3x and dx=1 into the equation and solve for dy. This gives us dy=cx+kyax+by. Substituting the values of x, y, and dx, we can find the values of a, b, c, and k.

For part (b), there is not just one correct answer. The set of choices for (a, b, c, k) that should be accepted is any set of numbers that satisfy the equation.

For part (c), we substitute the values of a, b, c, and k found in part (a) into the differential equation, and solve for y(2)=-2549 to find a solution satisfying the given condition.