Question

Sove the initial value problem dy/dx=12x3−2sinx,y(0)=3

Answer 1

Answer on Question #69663 – Math – Differential Equations

Question

Solve the initial value problem

$$\frac {d y}{d x} = 12x^3 - 2 \sin x,$$ $$y(0) = 3.$$

Solution

We can rewrite the equation (1) in the following form:

$$dy = (12x^3 - 2 \sin x)dx.$$

Now we shall integrate both sides of the previous equation:

$$\int dy = \int (12x^3 - 2 \sin x)dx,$$ $$y = y(x) = 3x^4 + 2 \cos x + C.$$

To find a constant $C$, we shall apply the initial condition (2) to the formula (3):

$$y(0) = 2 + C = 3,$$ $$C = 1.$$

Answer:

$$y(x) = 3x^4 + 2 \cos x + 1.$$

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