(a) Solve the following differential equation : dxdy=ex−y+x2e−y. OR (b) Solve the following differential equation : (x2−y2)dx+2xydy=0.
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Step-by-Step Solution
Step 1: Rewrite the differential equation
The given differential equation is dxdy=ex−y+x2e−y. We can rewrite ex−y as exe−y. This allows us to factor out e−y from the right-hand side.
Step 2: Separate the variables
Factor out e−y from the right-hand side. Then, separate the variables by multiplying both sides by ey and dx. This puts all terms involving y on one side and all terms involving x on the other, making it a separable differential equation.
Step 3: Integrate both sides
Now that the variables are separated, integrate both sides of the equation. The integral of ey with respect to y is ey, and the integral of ex+x2 with respect to x is ex+3x3. Remember to add the constant of integration.
Step 4: Solve for y
After integrating, we get ey=ex+3x3+C, where C is the constant of integration. This is the general solution to the differential equation.