Question

Which of the following represent the solution of the differential equation d^2y/dx^2+4y=0

5tan2x+5cos2x
5sin2x+4cos2x
5sin2x−3cos2x
5sin^22x−3cos2x

Answer 1

Answer on Question #62147 – Math – Differential Equations

Question

Which of the following represent the solution of the differential equation $d^2y/dx^2 + 4y = 0$

5tan2x+5cos2x

5sin2x+4cos2x

5sin2x-3cos2x

5sin^22x-3cos2x

Solution

The differential equation

$$\frac{d^2y}{dx^2} + 4y = 0$$

has the characteristic equation

$$\lambda^2 + 4 = 0,$$

its roots are

$$\lambda_1 = 2i, \lambda_2 = -2i.$$

Hence the solution of the differential equation (1) is

$$y = C_1 \sin(2x) + C_2 \cos(2x);$$

So, $y = 5\sin2x + 4\cos2x$ and $y = 5\sin2x - 3\cos2x$ can be solutions of the differential equation (1).

Answer: $y = 5\sin2x + 4\cos2x; y = 5\sin2x - 3\cos2x.$

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