What is the particular solution to the differential equation (dy)/(dx)=(4)/(x^(2)e^(2y)) with the initial condition y(1)=0 ?

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What is the particular solution to the differential equation  (dy)/(dx)=(4)/(x^(2)e^(2y)) with the initial condition  y(1)=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. We can do this by multiplying both sides by e^(2y) and dx, and dividing both sides by x^2. e^(2y) dy = (4/x^2) dxIntegrate both sides: Integrate both sides of the equation. We need to integrate e^(2y) with respect to y and 4/x^2 with respect to x. ∫ e^(2y) dy = ∫ (4/x^2) dxPerform integration: Perform the integration. The integral of e^(2y) with respect to y is (1/2)e^(2y), and the integral of 4/x^2 with respect to x is -4/x. (1/2)e^(2y) = -4/x + C, where C is the constant of integration.Solve for constant: Solve for the constant of integration using the initial condition. We are given that y(1) = 0. Plugging these values into our integrated equation gives us: (1/2)e^(2*0) = -4/1 + C (1/2)*1 = -4 + C C = 4 + 1/2 C = 4.5Write particular solution: Write the particular solution with the constant. Now that we have the value of C, we can write the particular solution: (1/2)e^(2y) = -4/x + 4.5Check with initial condition: Check the solution with the initial condition. Plugging x = 1 and y = 0 into the particular solution to see if both sides are equal: (1/2)e^(2*0) = -4/1 + 4.5 1/2 = -4 + 4.5 1/2 = 0.5 Both sides are equal, so the initial condition is satisfied.

What is the particular solution to the differential equation $\frac{dy}{dx}$ = $\frac{4}{x^{2}e^{2y}}$ with the initial condition $y(1)=0$ ?

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What is the particular solution to the differential equation $\frac{dy}{dx}$ = $\frac{4}{x^{2}e^{2y}}$ with the initial condition $y(1)=0$ ?