The given differential equation is (dy)/(dx) = e^(x - y) + x² e^(-y). We can rewrite e^(x-y) as e^x e^(-y). This allows us to factor out e^(-y) from the right-hand side.

(a) Solve the following differential equation : $\frac{dy}{dx} = e^{x - y} + x^2 e^{-y}$ . OR (b) Solve the following differential equation : $(x^2 - y^2) dx + 2xy dy = 0$ .

Answer: $e^y = e^x + \frac{x^3}{3} + C$

Step-by-step solution

Step 1: Rewrite the differential equation

The given differential equation is $\frac{dy}{dx} = e^{x - y} + x^2 e^{-y}$ . We can rewrite $e^{x-y}$ as $e^x e^{-y}$ . 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 $e^y$ 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 $e^y$ with respect to $y$ is $e^y$ , and the integral of $e^x + x^2$ with respect to $x$ is $e^x + \frac{x^3}{3}$ . Remember to add the constant of integration.

Step 4: Solve for y

After integrating, we get $e^y = e^x + \frac{x^3}{3} + C$ , where $C$ is the constant of integration. This is the general solution to the differential equation.

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(a) Solve the following differential equation : dydx=ex−y+x2e−y\frac{dy}{dx} = e^{x - y} + x^2 e^{-y}dxdy​=ex−y+x2e−y. OR (b) Solve the following differential equation : (x2−y2)dx+2xydy=0(x^2 - y^2) dx + 2xy dy = 0(x2−y2)dx+2xydy=0.