Solve the Equation d^2y/dx^2 + 4y = sin2x | Solve (D^2+4)y=sin2x

Here we solve (D^2+4)y=sin2x. The general solution of d²y/dx² + 4y = sin2x is given by y=c₁cos2x + c₂sin2x – (xcos2x)/4. In this page, let us learn to solve the equation d^2y/dx^2+4y=sin2x, i.e, (D^2+4)y=sin2x.

Solve (D²+4)y=sin2x

Question: Solve the equation $\dfrac{d^2y}{dx^2}$ + 4y = sin2x.

Solution:

The given differential equation can be written in the following form:

(D²+4)y = sin2x

where D² = d²/dx².

[Complementary Function (CF)]:

The auxiliary equation is given by

m²+4 = 0

⇒ m² = -4

⇒ m = $\pm \sqrt{-4}$

⇒ m = $\pm 2i$

The complementary function (CF) = c₁cos2x + c₂sin2x where c₁ and c₂ are arbitrary constants.

[Particular Integral (PI)]:

Replacing D² = -a² = -2², we have D²+4 = -2²+4 = 0.

So the usual well-known formula fails.

In this case, the particular integral is given by

$\dfrac{1}{D^2+4}\sin 2x$

= $x \dfrac{1}{\frac{d}{dD}(D^2+4)}\sin 2x$

= $x \dfrac{1}{2D}\sin 2x$

= $\dfrac{x}{2} \dfrac{1}{D}\sin 2x$

= $\dfrac{x}{2} \displaystyle \int \sin 2x ~dx$

= $\dfrac{x}{2} \dfrac{\cos 2x}{-2}$

= $-\dfrac{x\cos 2x}{4}$

∴ PI = $-\dfrac{x\cos 2x}{4}$.

General Solution: So the general solution of $\dfrac{d^2y}{dx^2} + 4y = \sin 2x$ is given by

y = CF + PI

⇒ y = c₁cos2x + c₂sin2x $- \dfrac{x \cos 2x}{4}$ where c₁ and c₂ are arbitrary constants of integration.

Also Study:

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