What will be the general solution of the differential equation d^2y/dx^2 = e^2x(12 cos3x – 5 sin3x)? (here, A and B are integration constant)

Category: QuestionsWhat will be the general solution of the differential equation d^2y/dx^2 = e^2x(12 cos3x – 5 sin3x)? (here, A and B are integration constant)
Editor">Editor Staff asked 11 months ago

What will be the general solution of the differential equation d^2y/dx^2 = e^2x(12 cos3x – 5 sin3x)? (here, A and B are integration constant)
 
(a) y = e^x sin3x + Ax + B
 
(b) y = e^2x sin3x + Ax + B
 
(c) y = e^2x sin3x + A
 
(d) Data inadequate
 
I had been asked this question at a job interview.
 
This question is from Linear Second Order Differential Equations in section Differential Equations of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct option is (b) y = e^2x sin3x + Ax + B
 
The explanation is: Given, d^2y/dx^2 = 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/2^2 + 3^2)[2cos3x + 3sin3x] – 5 * (e^2x/2^2 + 3^2)[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/2^2 + 3^2)[2cos3x + 3sin3x] + 2*(e^2x/2^2 + 3^2)[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