An ordinary differential equation is given below 6 {d^{2} y}/{d x^{2}}+\frac{d y}{d x}-y=0 The general solution of the above equation (with constants {C}_{1} and {C}_{2} ), is

Correct Answer is : y(x)=C1ex3+C2e−x2

y(x)=C_{1} x e^{\frac{x}{3}}+C_{2} e^{\frac{x}{2}}

y(x)=C_{1} e^{-\frac{x}{3}}+C_{2} x e^{\frac{x}{2}}

y(x)=C_{1} e^{-\frac{x}{3}}+C_{2} e^{\frac{x}{2}}

y(x)=C_{1} e^{\frac{x}{3}}+C_{2} e^{-\frac{x}{2}}

GATE Civil Engineering (CE) 2020 Shift 2 Solved Paper

Question : 58 of 65

Marks: +1, -0

An ordinary differential equation is given below

6 \;{d^{2} y}/{d x^{2}}+\;{d y}/{d x}-y=0

The general solution of the above equation (with constants

{C}_{1}

and

{C}_{2}

), is

Solution:

\;{6 d^{2} y}/{d x^{2}}+\;{d y}/{d x}-y=0

(6 m^{2}+m-1) y=0

m=\;{d}/{d x}

Auxiliary equation is

6 m^{2}+m-1=0

6 m^{2}+3 m-2 m-1=0

3 m(2 m+1)-1(2 m+1)=0

(2 m+1)(3 m-1)=0

m=-\;{1}/{2}, \;{1}/{3}

{CF}_{.}= {C}_{1} {e}^{-\;{1}/{2} {x}}+ {C}_{2} {e}^{\;{1}/{3} {x}}

{PI}=0

{f}( {x})=0

General solution

{y}= {C}

_1e^{-x {/} 2}+C_{2} e^{x {/ }3}

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