GATE ECE 2020 | Question: 4

Question

The general solution of $\dfrac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}-6\dfrac{\mathrm{d} y}{\mathrm{d} x}+9y=0$ is

• A. $y=C_{1}e^{3x}+C_{2}e^{-3x}$
• B. $y=(C_{1}+C_{2}x)e^{-3x}$
• C. $y=(C_{1}+C_{2}x)e^{3x}$
• D. $y=C_{1}e^{3x}$

Answer 1

Since, This is a linear differential equation of the form: d²y/dx² + Pdy/dx + Qy = R. 

Here P= -6, Q=9, R=0 

Now, Converting this in Auxiliary Equation; 

           m² - 6m + 9 = 0

                             m = 3,3

Since, the roots of this eqaution are equal;

Therefore, CF= (C₁ + C₂x)e^mx  i.e

                  CF= (C₁ + C₂x)e^3x 

and R=0 ; So, PI=0;

Hence, General Solution= CF+PI

               i.e  general solution =  (C₁ + C₂x)e^3x