What is the particular solution to the differential equation (dy)/(dx)=(3+y)/(1-2x) with the initial condition y(0)=0 ?

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What is the particular solution to the differential equation  (dy)/(dx)=(3+y)/(1-2x) with the initial condition  y(0)=0 ?

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Separate variables: Separate the variables in the differential equation. We want to get all the y terms on one side and all the x terms on the other side. This can be done by multiplying both sides by (1-2x)dx and dividing by (3+y). So we get (1-2x)dx = (dy)/(3+y).Integrate both sides: Integrate both sides of the equation. We integrate the left side with respect to x and the right side with respect to y. ∫(1-2x)dx = ∫(dy)/(3+y)Perform integration: Perform the integration. The left side becomes x - x^2 + C1, where C1 is the constant of integration. The right side becomes ln|3+y| + C2, where C2 is the constant of integration. So we have x - x^2 + C1 = ln|3+y| + C2.Combine constants: Combine the constants of integration. Since C1 and C2 are both constants, we can combine them into a single constant, which we'll call C. So we have x - x^2 = ln|3+y| + C.Apply initial condition: Apply the initial condition to find C. We are given that y(0) = 0, so we plug in x = 0 and y = 0 into the equation. 0 - 0^2 = ln|3+0| + C This simplifies to 0 = ln(3) + C. So C = -ln(3).Write particular solution: Write the particular solution with the value of C. Substitute C back into the equation to get the particular solution. x - x^2 = ln|3+y| - ln(3)Simplify equation: Simplify the equation by using properties of logarithms. We can use the property of logarithms that ln(a) - ln(b) = ln(a/b) to combine the logarithms on the right side. x - x^2 = ln(|3+y|/3)Remove absolute value: Remove the absolute value. Since ln(a) is only defined for a > 0, and 3+y > 0 for the solution to exist, we can remove the absolute value. x - x^2 = ln((3+y)/3)Exponentiate to solve: Exponentiate both sides to solve for y. Raise e to the power of both sides to get rid of the natural logarithm. e^(x - x^2) = (3+y)/3Solve for y: Solve for y. Multiply both sides by 3 to isolate y. 3e^(x - x^2) = 3+y y = 3e^(x - x^2) - 3

What is the particular solution to the differential equation $\frac{dy}{dx}=\frac{3+y}{1-2x}$ with the initial condition $y(0)=0$ ?

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What is the particular solution to the differential equation $\frac{dy}{dx}=\frac{3+y}{1-2x}$ with the initial condition $y(0)=0$ ?