The particular solution for the differential equaiton d2ydx2-3dydx-2y-5 cos x is

Given DE is d^2ydx^2+3dydx+2y=5 cos x P.I=1(D^2+3D+2).5 cos x Using =1f(D^2)cos(ax+b)=1f( a^2)cos(ax+b) =P.I=51( 1^2)+3D+2 cos x =5(3D+1) cos x =5×3D 19D^2 1 co ...

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Engineering Mathematics

Non-Homogeneous Linear Differential Equations (Methods for finding PI for sin(ax) ,cos(ax), x^m)

The particula...

Question

The particular solution for the differential equaiton $\frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + 2 y = 5 c o s x$ is

0.5 c o s x + 1.5 s i n x

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1.5 c o s x + 0.5 s i n x

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1.5 s i n x

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0.5 c o s x

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Solution

The correct option is A $0.5 c o s x + 1.5 s i n x$
Given DE is
$\frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + 2 y = 5 c o s x$
$P . I = \frac{1}{(D^{2} + 3 D + 2)} .5 c o s x$
Using
$= \frac{1}{f (D^{2})} c o s (a x + b) = \frac{1}{f (- a^{2})} c o s (a x + b)$
$= P . I = 5 \frac{1}{(- 1^{2}) + 3 D + 2} c o s x$
$= \frac{5}{(3 D + 1)} c o s x$
$= 5 \times \frac{3 D - 1}{9 D^{2} - 1} c o s x$
$= 5 \times \frac{3 D - 1}{9 (- 1) - 1} c o s x$
$= \frac{5}{- 10} \times (3 D - 1) c o s x$
$P . I = - \frac{1}{2} \times (3 D - 1) c o s x$
$= - \frac{1}{2} \times [(3 (- s i n x) - c o s x]$
$P . I = 1.5 s i n x + 0.5 c o s x$

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The particular solution for the differential equaiton d2ydx2+3dydx+2y=5cosx is

The particular solution for the differential equaiton $\frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} + 2 y = 5 c o s x$ is