ODE 1st orderMedium

Separable variable ODE

Solve the ODE y' = 2xy with initial condition y(0) = 3.

Given data

y' = 2xyy(0) = 3

Steps

0 / 3

Separate the variables: move all y terms to the left and x terms to the right.
Write your working — the numerical result must appear in the text (logica)
Integrate both sides.
Apply the initial condition y(0) = 3 and find the explicit solution.

Full worked solution

Separate the variables: move all y terms to the left and x terms to the right.
$\frac{dy}{y} = 2x\,dx$
$\frac{dy}{dx} = 2xy \rightarrow \frac{dy}{y} = 2x\,dx$ . Separating variables collects $y$ on the left and $x$ on the right.
Integrate both sides.
$\ln|y| = x^2 + C$
$\int \frac{dy}{y} = \int 2x\,dx \rightarrow \ln|y| = x^2 + C$ . Integrating both sides yields the general solution in implicit form.
Apply the initial condition y(0) = 3 and find the explicit solution.
$y(0)=3 \Rightarrow C=\ln3 \Rightarrow y = 3e^{x^2}$
$y = e^{x^2 + C} = e^C\,e^{x^2}$ . Using $y(0) = 3$ : $3 = e^C\cdot e^0 = e^C \rightarrow K = 3$ . Explicit solution: $\mathbf{y(x) = 3\,e^{x^2}}$ .

Result:y(x) = 3e^{x²}.

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