Recurrence Equation Solution / Difference equation - WolframAlpha

Question

I am given the recurrece equation

$y_k-7y_{k-1}=5^k$

and found the (hopefully correct) particular solution to be

$y_k^P=-\frac{5}{2}5^k$

WolframAlpha, however, gives the particular solution

$y_k^P=c_17^{k-1}+\frac{5}{2}(7^k-5^k)$

So my question is: How can I rewrite WA's equation to my equation? For the life of me I cannot do it...

Answer 1

$y_k^P=- \frac{5}{2}5^k$: $- \frac{5}{2}5^k - 7(- \frac{5}{2}5^{k-1})=5^k$ is a particular solution, but the General solution is the Superposition of the particulary and the homogenous solution $y_k^H$, i.e. $y_k = y_k^P+y_k^H$. The homogenous solution can be obtained from the equation $y_k^H-7y_{k-1}^H=0$. This equation can be solved to give $y_k^H=c 7^k = (c_1 + \frac{5}{2}) 7^k$; for arbitrary constant $c$.