this question is posted twice to let the best answer get 3000 points by posting

Question

this question is posted twice to let the best answer get 3000 points by posting his answer twice in both of them .. clear expalined solutions with steps will accepted only .

Consider the following initial value problem: dy (x y+ 202 dx y (0) 3-2 1. Write the equation in the form dy G (ax by c dx where a, b, and c are constants and G is a function. 2. Use the substitution u ax by c to transfer the equation into the variables u and x only 3. Solve the equation in (2) 4. Re-substitute u ax by c to write your solution in terms of x and y. 5. Use the initial condition to write the solution for this initial value problem.

Explanation / Answer

1) Here, dy/dx = (x+y+2)^2 = (x+y+2)(x+y+2), Let ax+by+c = x+y+2 , so a=1,b=1,c=2, G = x+y+2.

2) Let u = x+y+2, du/dx = 1+ dy/dx.

dy/dx = du/dx - 1. Since, (x+y+2)^2 = u^2, the equation in u and x is du/dx = u^2 +1.

3) Solving du/dx = u^2 +1, we have x = integral of (1/(u^2 +1)), So x=atan(u) + c or u = tan(x) + C.

4) Resubstituting u=x+y+2, we have x+y+2 = tan(x) + C.

5) Using the initial value problem y(0) = -2, we have C = 0, then the final equation is x+y+2 = tan(x)

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.