Question

Find the particular solution of d2y/d x2 +4. dx/Dy +4y= 2 cos²x

Answer 1

The solution to the equation are as follows.

y = c₁$e^{\sqrt{-3 -sin²(x)x} }$ + c₂                $e^{\sqrt{-3 -sin²(x)x} }$

What is differential equation?

In arithmetic, a equation is associate equation that relates one or a lot of unknown functions and their derivatives.

Main body:

Find the first derivative.

dy/dx=r$e^{rx}$

Find the second derivative.

d²y/dx² =r² $e^{rx}$

Substitute into the differential equation.

r²$e^{rx}$+4$e^{rx}$=cos ²x

Factor $e^{rx}$out of r²$e^{rx}$.

$e^{rx}$r²+4$e^{rx}$=cos²x

Factor $e^{rx}$ out of 4$e^{rx}$.

$e^{rx}$r²+$e^{rx}$⋅4=cos²x

Factor out of $e^{rx}$r²+4$e^{rx}$

$e^{rx}$(r²+4)=cos²x

Since exponentials can never be zero, divide both sides by

$e^{rx}$r²+4=cos²x

Use the double-angle identity to transform

cos²x to 1−sin²(x).

r² +4=1−sin²(x)

Subtract 4 from both sides of the equation.

r²=1−sin²(x)−4

Simplify the right side.

r² = -3 -sin²x

r = √-3 -sin²x

By the principle of superposition, the general solution is a linear combination of the two solutions for a second order homogeneous linear differential equation.

y = c₁$e^{\sqrt{-3 -sin²(x)x} }$ + c₂

Hence the differential solution is as follows.

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