My Coding > Mathematics > First order ordinary differential equations > Linear first order ODE

Linear first order ODE (Page: 2)

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  1. Integration factor;
  2. Solving \(y\prime = 2xy\) with Integrating factor;
  3. Solving \(y\prime = 2xy\) by separating variables;

Let's try to solve a simple equation with an integration factor. As an example, we can use a very simple linear differential equation \(y\prime = 2xy\). This is our formula for solving linear ODE:

\[\boxed{y\prime + P(x)y = Q(x) \\ y = \frac{1}{I}\int IQ\mathop{dx} + \frac{C}{I}; I = e^{\int P\mathop{dx}}}\]

Because we just derived this equation, I will use it. Otherwise, I do advise you to derive it every time with your real data.

\[y\prime = 2xy; P = -2x; Q = 0\]

\[I = e^{\int 2xmathop{dx}} = e^{-x^2 + C} = Ae^{-x^2} = e^{-x^2}\]

Now we can apply this integration factor and solve this equiation:

\[y = \frac{C}{e^{-x^2}} = Ce^{x^2}\]

This solution is shown in this video:

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Published: 2023-05-14 01:12:52
Updated: 2023-05-14 01:39:32

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