Let $1 ,x$ and $x^2$ be the solution of a second order linear Non homogenous differential equation on $-1 < x < 1$, then it's general solution involving arbitrary constants can be written as :
(a) $c_1(1-x) + c_2(x – x^2) +1$
(b) $c_1(x) + c_2 ( x^2) +1$
(c) $c_1(1+x) + c_2(1 + x^2) +1$
(d) $c_1 + c_2 x + x^2$
Now, I know this : The general solution of such a differential equation is written as:
$Y = c_1 f + c_2 g + \text{P.I.}$
where $f$ and $g$ are two Linearly Independent solutions and $P.I.$ denotes the particular integral obtained by solving the non homogeneous part.
So, Using this fact I know that options (b) and (c) are false because the function are linearly Dependent on given interval.
However I am confused between (a) and (d) .The given functions are Linearly Independent but I have no idea how to decide the Particular Integral.
Can anyone tell me how should I tackle options (a) and (d) ?
Thank you.
Best Answer
For the d option. $$y(x)=c_1+c_2x+x^2$$ substitute $x=e^t$ $$y(t)=c_1+c_2e^t+e^{2t}$$ $$r=0, r=1 \implies r(r-1)=0 \implies y''-y'=0$$ $$y''-y'=f(x)$$ Particular solution is $e^{2t}$ $$4e^{2t}-2e^{2t}=f(x) \implies f(x)=2e^{2t}$$ The equation is $$y''-y'=2e^{2t}$$ $$\implies x^2y(x)''=2x^2 \implies y''(x)=2$$
And $y''=2$ has option d as solution. Integrate it.
For option $(b)$ I got the equation
$$y''(t)-3y'(t)+2y(t)=2$$ It gives Cauchy-Euler's equation: $$\implies x^2y''(x)-2xy'(x)+2y(x)=2$$ Has solution: $$y(x)=c_1x^2+c_2x+1$$