29. (a) Find the general solution of the differential equation
2x2dxdy=y2+2xy.
(differential equation, general solution, calculus, homogeneous equation)
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Step-by-Step Solution
Step 1: Rearrange the differential equation
First, we rearrange the given differential equation to express dxdy in terms of x and y. This helps us identify the type of differential equation we are dealing with. Dividing both sides by 2x2 gives us the standard form.
Step 2: Check for homogeneity and substitute
This is a homogeneous differential equation because all terms in the numerator and denominator have the same degree (degree 2). To solve it, we use the substitution y=vx, which implies dxdy=v+xdxdv. We substitute y=vx into the equation and simplify.
Step 3: Separate variables
Now we substitute v+xdxdv for dxdy and rearrange the equation to separate the variables v and x. We move v to the right side, combine terms, and then isolate dv and dx on opposite sides of the equation.
Step 4: Integrate both sides
We integrate both sides of the separated equation. The integral of v−2 is −v−1, and the integral of x−1 is ln∣x∣. We add the constant of integration, C, on one side.
Step 5: Substitute back for v and simplify
Finally, we substitute back v=xy into the integrated equation to express the solution in terms of y and x. We can also rewrite the constant −C as a new constant C1 for simplicity and solve for y.