Solve the differential equation 𝑑 𝑦 𝑑 𝑥 = 2 𝑥 dx dy =2x with initial condition 𝑦 ( 0 ) = 3 y(0)=3.

Question

Solve the differential equation

$\frac{dy}{dx} = 2x$ with initial condition $y(0) = 3.$

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Solution

Sure, let's solve the differential equation dy/dx = 2x with the initial condition y(0) = 3.

Step 1: Separate the variables. We can rewrite the differential equation as dy = 2x dx.

Step 2: Integrate both sides. The integral of dy is y, and the integral of 2x dx is x^2. So, we have y = x^2 + C, where C is the constant of integration.

Step 3: Apply the initial condition. We know that y(0) = 3, so we can substitute these values into the equation to solve for C. We get 3 = 0^2 + C, so C = 3.

Therefore, the solution to the differential equation is y = x^2 + 3.

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Solve the differential equation 𝑑 𝑦 𝑑 𝑥 = 2 𝑥 dx dy ​ =2x with initial condition 𝑦 ( 0 ) = 3 y(0)=3.

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