What will be the general solution of the differential equation d2y/dx2= e2x(12 cos3x – 5 sin3x)? (here, A and B are integration constant)a)y = exsin3x + Ax + Bb)y = e2xsin3x + Ax + Bc)y = e2xsin3x + Ad)Data inadequateCorrect answer is option 'B'. Can you explain this answer?

Question

What will be the general solution of the differential equation d2y/dx2= e2x(12 cos3x &ndash; 5 sin3x)? (here, A and B are integration constant)a)y = exsin3x + Ax + Bb)y = e2xsin3x + Ax + Bc)y = e2xsin3x + Ad)Data inadequateCorrect answer is option 'B'. Can you explain this answer?

Answer

The general solution of the differential equation d2y/dx2= e2x(12 cos3x) is:

y(x) = Ae2x + Be-x + Csin3x + Dcos3x + (1/72)e2xsin3x - (1/216)e2xcos3x

where A, B, C, and D are constants of integration.

Answer

Given, d²y/dx² = e^2x(12 cos3x – 5 sin3x) ………(1)
Integrating (1) we get,
dy/dx = 12∫ e^2xcos3x dx – 5∫ e^2x sin3x dx
= 12 * (e^2x/2² + 3²)[2cos3x + 3sin3x] – 5 * (e^2x/2² + 3²)[2sin3x – 3cos3x] + A (A is integrationconstant)
So dy/dx = e^2x/13 [24 cos3x + 36 sin3x – 10 sin3x + 15 cos3x] + A
= e^2x/13(39 cos3x + 26 sin3x) + A
⇒ dy/dx = e^2x(3 cos3x + 2 sin3x) + A ……….(2)
Again integrating (2) we get,
y = 3*∫ e^2x cos3xdx + 2∫ e^2x sin3xdx + A ∫dx
y = 3*(e^2x/2² + 3²)[2cos3x + 3sin3x] + 2*(e^2x/2² + 3²)[2sin3x – 3cos3x] + Ax + B (B is integration constant)
y = e^2x/13(6 cos3x + 9 sin3x + 4 sin3x – 6 cos3x) + Ax + B
or, y = e^2x sin3x + Ax + B

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