Question

PPlease answer using MATLAB

Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. day - 6 dy + y = 0; y = C1e3x + czxe3x dx2 dx when y = Gjer Czxe3x dy dx dey dx2 = Thus, in terms of x, d2y + 9 = dx2 dx - dy +91c1e3x + cqxe3x)

Answer 1

The given diffential equation is 
$\frac{d^2y}{dx^2}-6\frac{dy}{dx}+9y=0$,   
y = Cier + C2.ce
.

$\text{Now we find}\quad \frac{dy}{dx} \quad \text{and} \quad \frac{d^2y}{dx^2}\quad \text{of}\quad y=c_1e^{3x}+c_2xe^{3x},\quad \text{by using matlab}$Matlab code single derivative
syms x c1 c2
diff(c1*exp(3x)+c2*x*exp(3x),x)

Ans=

3*c1*exp(3x)+c2*exp(3x)+3*c2*x*exp(3x)

that is

$\frac{dy}{dx}=3c_1e^{3x}+c_2e^{3x}+3c_2xe^{3x}$

Matlab code double derivative

syms x c1 c2
diff(diff(c1*exp(3x)+c2*x*exp(3x),x))

Ans=

9*c1*exp(3x)+6*c2*exp(3x)+9*c2*x*exp(3x)

that is

$\frac{d^2y}{dx^2}=9c_1e^{3x}+6c_2e^{3x}+9c_2xe^{3x}$

The given differential equation in term of x, is

$\frac{d^2y}{dx^2}-6 \frac{dy}{dx}+9y=9c_1e^{3x}+6c_2e^{3x}+9c_2xe^{3x}-6(3c_1e^{3x}+c_2e^{3x}+3c_2xe^{3x})+{\color{Red} 9(c_1e^{3x}+c_2xe^{3x})}\\=0$