ODEMedium

Differential equation — separable variables

Solve the differential equation y' = 2x·y with initial condition y(0) = 3.

Given data

y' = 2x·yy(0) = 3

Steps

0 / 4

Separate the variables: move y to the left and x to the right.
Write your working — the numerical result must appear in the text (logica)
Integrate both sides: ∫ dy/y = ∫ 2x dx.
Apply the exponential to isolate y.
Use the condition y(0) = 3 to find K.

Full worked solution

Separate the variables: move y to the left and x to the right.
$\dfrac{dy}{y} = 2x\,dx$
$\frac{dy}{dx} = 2xy \rightarrow \frac{dy}{y} = 2x\,dx$ . Separating variables puts all $y$ terms on the left and all $x$ terms on the right.
Integrate both sides: ∫ dy/y = ∫ 2x dx.
$\ln|y| = x^2 + C$
$\int \frac{dy}{y} = \int 2x\,dx \rightarrow \ln|y| = x^2 + C$ . Integrating both sides yields the general solution in implicit form.
Apply the exponential to isolate y.
$|y| = e^{x^2 + C} = e^C \cdot e^{x^2}$
$|y| = e^{x^2 + C} = e^C\,e^{x^2} \rightarrow y = K\,e^{x^2}$ with $K = \pm e^C$ . Exponentiating both sides removes the logarithm.
Use the condition y(0) = 3 to find K.
$y(0) = K \cdot e^0 = K = 3$
$3 = K\,e^{0} = K$ , so $\mathbf{K = 3}$ . The particular solution satisfying the initial condition is $\mathbf{y(x) = 3\,e^{x^2}}$ .

Result:y(x) = 3·e^{x²}.

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