$\mathrm{p}$ and $\mathrm{q}$ are positive integers and $\mathrm{n} < \mathrm{r} < \mathrm{m}$. If the order…
$\mathrm{p}$ and $\mathrm{q}$ are positive integers and $\mathrm{n} < \mathrm{r} < \mathrm{m}$. If the order and degree of the differential equation $\left(\frac{d^m y}{d x^m}+\frac{d^n y}{d x^n}\right)^{p / q}=5 \frac{d^r y}{d x^r}$ are respectively 4 and 3 , then
$\mathrm{n}=4, \mathrm{q}=3$
$m=4, q=3$
$r=4, q=3$
$m=4, p=3$
Solution
Given
$
\left(\frac{d^m y}{d_x m}+\frac{d^n y}{d_x n}\right)^{\frac{p}{q}}=5 \frac{d^x y}{d_x r}
$
with order 4 and degree 3 .
$
\left(\frac{d^m y}{d_x m}+\frac{d^x y}{d_x n}\right)=5^q\left(\frac{d^r y}{d_x r}\right)^q
$
Here, $m>r>x$. there $\frac{d^m y}{d_x n}$ will be the higher order operator
$
\text { So, } \mathrm{m}=4
$
Now, $\frac{d^m y}{d_x m}$ has a power 3 .
Then, $\mathrm{P}=3$