he particular solution of the differential equation (dy)/(dx)-e^(x)=ye^(x), when x=0 and y=1 is

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he particular solution of the differential equation  (dy)/(dx)-e^(x)=ye^(x), when  x=0 and  y=1 is

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Rewrite Differential Equation: First, we need to rewrite the differential equation in a form that allows us to use the method of integrating factors. The equation is already in the form of a first-order linear differential equation, dy/dx + P(x)y = Q(x), where P(x) = -e^x and Q(x) = e^x.Calculate Integrating Factor: Next, we calculate the integrating factor, μ(x), which is e^(∫P(x)dx). In this case, μ(x) = e^(∫-e^xdx). The integral of -e^x with respect to x is -e^x, so μ(x) = e^(-e^x).Multiply by Integrating Factor: Now, we multiply the entire differential equation by the integrating factor, μ(x), to get μ(x) * (dy/dx) + μ(x) * P(x) * y = μ(x) * Q(x). Substituting the values of μ(x), P(x), and Q(x), we get e^(-e^x) * (dy/dx) - e^(-e^x) * e^x * y = e^(-e^x) * e^x.Recognize Derivative Form: We notice that the left side of the equation is now the derivative of (y * μ(x)) with respect to x. Therefore, we can write d/dx(y * e^(-e^x)) = e^(-e^x) * e^x.Integrate Both Sides: To solve for y, we integrate both sides of the equation with respect to x. The integral of the left side is y * e^(-e^x), and the integral of the right side is ∫e^(-e^x) * e^x dx.Re-evaluate Integrating Factor: The integral on the right side is not a standard integral and cannot be expressed in terms of elementary functions. It seems there has been a mistake in the calculation of the integrating factor or in the setup of the equation. We need to re-evaluate the integrating factor and the setup of the equation.

The particular solution of the differential equation $\frac{dy}{dx}-e^{x}=ye^{x}$ , when $x=0$ and $y=1$ is

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The particular solution of the differential equation dydx−ex=yex\frac{dy}{dx}-e^{x}=ye^{x}dxdy​−ex=yex, when x=0x=0x=0 and y=1y=1y=1 is

The particular solution of the differential equation $\frac{dy}{dx}-e^{x}=ye^{x}$ , when $x=0$ and $y=1$ is