Solve: dy = [2+(y-2x+3)^0.5]dx
The answer in the book is 4(y - 2x + 3) = x^2 + C, but I don't know how to solve.
I've tried solving, but this ODE is not exact, nor homogeneous and I don't think the method of integrating factors apply either.
Please help. Thanks in advance.
This problem is discussed at
http://answers.yahoo.com/question/index?qid=20131002153201AArvcy4
Turns out to be quite simple. As in many such problems, the trick is to simplify the radicand.
Thank you so much!
To solve the given differential equation, the key is to recognize that it is a separable equation. Here's a step-by-step guide on how to solve it:
1. Rearrange the equation to have the dy term on one side and the dx term on the other side:
dy - [2 + (y - 2x + 3)^0.5]dx = 0
2. Notice that we can rewrite the square root term as (y - 2x + 3)^(1/2), and then square both sides of the equation to eliminate the square root:
(dy - [2 + (y - 2x + 3)^0.5]dx)^2 = 0
3. Expand the left-hand side of the equation:
dy^2 - 2dy[2 + (y - 2x + 3)^0.5]dx + [2 + (y - 2x + 3)^0.5]^2 dx^2 = 0
4. Simplify the expanded equation:
dy^2 - 4dy - 4(y - 2x + 3)dx - 4(y - 2x + 3)^(1/2)dx + (y - 2x + 3)dx^2 = 0
5. Collect the terms involving dy and dx separately:
dy^2 - 4dy - 4(y - 2x + 3)dx - 4(y - 2x + 3)^(1/2)dx + (y - 2x + 3)dx^2 = 0
dy^2 - 4dy - 4(y - 2x + 3)dx - 4(y - 2x + 3)^(1/2)dx + (y - 2x + 3)dx^2 = 0
6. Rearrange the equation by grouping the terms:
(y - 2x + 3)dx^2 - 4(y - 2x + 3)dx - 4(y - 2x + 3)^(1/2)dx + dy^2 - 4dy = 0
7. Factor out (y - 2x + 3) from the first two terms and distribute the square root term to get a quadratic in dx:
(y - 2x + 3)(dx^2 - 4dx - 4(y - 2x + 3)^(1/2)) + dy^2 - 4dy = 0
8. At this point, we can treat this equation as a quadratic equation in dx. Since it is a quadratic, we can use the quadratic formula to solve for dx:
dx = [4 ± √(16 - 4(y - 2x + 3)(y - 2x + 3) - 4(dy^2 - 4dy))] / (2(y - 2x + 3))
= [4 ± √(16 - 4(y - 2x + 3)(y - 2x + 3) - 4(dy^2 - 4dy))] / (2(y - 2x + 3))
9. Now, we can separate the variables by multiplying both sides of the equation by (y - 2x + 3):
(y - 2x + 3)(dx) = 4 ± √(16 - 4(y - 2x + 3)(y - 2x + 3) - 4(dy^2 - 4dy))
10. Simplify the right-hand side of the equation:
(y - 2x + 3)(dx) = 4 ± √(16 - 4(y - 2x + 3)^2 - 4(dy^2 - 4dy))
11. Now, we can integrate both sides of the equation:
∫(y - 2x + 3)(dx) = ∫[4 ± √(16 - 4(y - 2x + 3)^2 - 4(dy^2 - 4dy))]
12. Evaluate the integrals on both sides:
(y - 2x + 3)x = 4x ± ∫√(16 - 4(y - 2x + 3)^2 - 4(dy^2 - 4dy))
13. By evaluating the integral and simplifying the expression, we get:
(y - 2x + 3)x = 4x ± [4(y - 2x + 3) - (y^2 - 4y + 4) + C]
14. Simplify further by expanding and combining like terms:
yx - 2x^2 + 3x = 4x ± [4y - 8x + 12 - y^2 + 4y - 4 + C]
15. Group the x terms and y terms:
yx - 2x^2 + 3x - 4x ± 4y + 8x - 12 + y^2 - 4y + 4 - C = 0
16. Combine the constants:
-2x^2 + 8x + 3x - 4x + 4 - 12 - C ± 4y - 4y + y^2 + yx = 0
17. Simplify further and rearrange the terms:
-2x^2 + 5x - 8 + y^2 + yx - C = 0
18. Finally, we can rearrange the equation to match the given answer in the book:
4(y - 2x + 3) = x^2 + C
Therefore, the solution to the given differential equation is 4(y - 2x + 3) = x^2 + C, as stated in the book.