Differentiate (dy)/(dx)=e^(x+y) with initial condition y(0)=-ln 3

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Differentiate  (dy)/(dx)=e^(x+y) with initial condition  y(0)=-ln 3

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Write Differential Equation: Write down the given differential equation and initial condition. (dy)/(dx) = e^(x+y) Initial condition: y(0) = -ln(3)Separate Variables: To solve the differential equation, we need to separate the variables x and y. We can do this by dividing both sides by e^y and multiplying both sides by dx. (dy)/(e^y) = e^x dxIntegrate Equations: Integrate both sides of the equation with respect to their respective variables. ∫(1/e^y) dy = ∫e^x dxApply Initial Condition: Perform the integration on both sides. The integral of 1/e^y with respect to y is -1/e^y, and the integral of e^x with respect to x is e^x. -1/e^y = e^x + C, where C is the constant of integration.Simplify Equation: Apply the initial condition to find the constant of integration C. y(0) = -ln(3) implies that when x = 0, y = -ln(3). Substitute x = 0 and y = -ln(3) into the integrated equation. -1/e^(-ln(3)) = e^0 + CFind Constant: Simplify the equation using the properties of exponents and logarithms. e^ln(3) = 3, so -1/3 = 1 + C.General Solution: Solve for C. C = -1/3 - 1 C = -4/3Solve for y: Write the general solution of the differential equation with the constant C. -1/e^y = e^x - 4/3Check for Errors: To express y explicitly in terms of x, we need to solve for y. e^y = -3/(e^x - 4/3) Taking the natural logarithm of both sides gives us y. y = ln(-3/(e^x - 4/3))Check for any mathematical errors in the previous steps, especially in the algebraic manipulations and the application of the initial condition.

Differentiate $\frac{d y}{d x}=e^{x+y}$ with initial condition $y(0)=-\ln 3$

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Differentiate $\frac{d y}{d x}=e^{x+y}$ with initial condition $y(0)=-\ln 3$