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bearhunter [10]
3 years ago
8

Find a solution to the initial value problem, y′′+18x=0,y(0)=5,y′(0)=1.

Mathematics
1 answer:
Serga [27]3 years ago
7 0

We want to find a solution to the initial value problem:

y'' + 18x = 0 \qquad,\qquad y(0) = 5 \qquad,\qquad y'(0)=1.

We can start by integrating the equation once:

\dfrac{\textrm{d}^2 y}{\textrm{d}x^2} + 18 x = 0 \iff \dfrac{\textrm{d}^2 y}{\textrm{d}x^2} = -18 x \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -18\displaystyle\int x\textrm{ d}x \iff \dfrac{\textrm{d}y}{\textrm{d}x}=-18\dfrac{x^2}{2} + C \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + C.

Using the initial condition y'(0) = 1, we can determine the integration constant C:

\dfrac{\textrm{d}y}{\textrm{d}x}\Big\vert_{x= 0} = 1 \iff -9 \times 0^2 + C = 1 \iff C = 1.

Therefore, we have:

\dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + 1

We can now integrate again:

y(x) = \displaystyle\int\dfrac{\textrm{d}y}{\textrm{d}x}\textrm{ d}x = \int\left(-9x^2+1\right)\textrm{d}x = -9\int x^2\textrm{ d}x + \int\textrm{d}x =\\\\= -9\dfrac{x^3}{3} + x + K = -3x^3 + x + K.

The integration constant K is determined by using y(0) = 5:

y(0) = 5 \iff -3 \times 0^3 + 0 + K = 5 \iff K = 5.

Finally, the solution is:

\boxed{y(x) = -3x^3 + x + 5}.

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