# Solve Second Order ODE using variation of parameters

Find the general solution for $t > 0$ using variation of parameters (look for fundamental set of homogeneous solutions $t^r$

$$t^2 x'' + 7tx' + 5x = t$$

• Have you started doing this task on your own? – svavil Nov 7 '15 at 22:00
• I tried solving the homogeneous equation but I don't know how to solve it with non-constant coefficients – user286826 Nov 7 '15 at 22:20

For the homogeneous solutions, you need to solve

$$t^2x''+7tx'+5x=0$$

Put $x=t^r$ with $r$ a fixed constant to be determined, then $x''=r(r-1)x^{r-2}$ and $x'=rx^{r-1}$.

Now you should b able to substitute into the homogeneous D.E. to get an equation involving $x^r$ on the left-hand side and zero on the right-hand side. Hence get an equation for the coefficient of $x^r$ and get a quadratic equation for $r$.

Note that the original D.E. has a particular solution $x_p=\frac{1}{12}t$

• If $t=x^r$ then $x''=r(r-1)x^{r-2}$ and $x'=rx^{r-1}$ and what is $x$? I would like to better understand how to solve such DFE's. – zoli Nov 7 '15 at 23:24
• @zoli - $x=x(t)$ is simply a function of $t$. I can see why you were confused - for some reason I wrote $t=x^r$ when meaning $x=t^r$ (as per the question). This is the hint in the question, and arises because the form of the homogeneous D.E. suggests a polynomial solution, i.e. for every successive derivative, the coefficient is one degree higher. – Marconius Nov 8 '15 at 0:45
• So edited you. Thank you. – zoli Nov 8 '15 at 0:47