Question

Solve the initial value problem d y d x = 2 x 1 , y ( 0 ) = 2

Answer 1

Using separation of variables, it is found that the solution to the initial value problem is of y(x) = x² + 2.

What is separation of variables?

In separation of variables, we place all the factors of y on one side of the equation with dy, all the factors of x on the other side with dx, and integrate both sides.

In this problem, the differential equation is given by:

$\frac{dy}{dx} = 2x$

Then, applying separation of variables:

$dy = 2x dx$

$\int dy = \int 2x dx$

$y = x^2 + K$

Since y(0) = 2, we have that the constant of integration is K = 2, and the solution is:

y(x) = x² + 2.

More can be learned about separation of variables at https://brainly.com/question/14318343

Answer 2

The differential equation is y(x) = x² + 2.

What is the differential equation?

Differential Equations In Mathematics, a differential equation is an equation that contains one or more functions with their derivatives.

The given equation is;

$\rm \dfrac{dy}{dx}=2x$

Applying the variable separation method;

$\rm \dfrac{dy}{dx}=2x\\\\\int\limits \, dy=\int\limits\, 2x. dx\\\\y = 2 \times \dfrac{x^{1+1}}{1+1} +c\\\\y = 2 \times \dfrac{x^{2}}{2} +c\\\\y = x^2+c$

The value of c when y( 0 ) = 2 is c =2.

Hence, the required differential equation is y(x) = x² + 2.

More can be learned about the differential equation at;

brainly.com/question/14318343

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