$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$
$\frac{\partial}{\partial t }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$
$\frac{1}{2\sqrt{1+t} }e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_{k+1}(x)}{z}\frac{t^{k}}{{k}!}$
$\frac{\partial}{\partial x }(\frac{1}{2\sqrt{1+t} }e^{x\sqrt{1+t}})=\frac{\partial}{\partial x }(\sum \limits_{k=0}^\infty \frac{U_{k+1}(x)}{x}\frac{t^{k}}{{k}!})$
$\frac{e^{x\sqrt{1+t}}}{2}=\sum \limits_{k=0}^\infty \frac{\partial}{\partial x }(\frac{U_{k+1}(x)}{x})\frac{t^{k}}{{k}!}$
$\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}=2\sum \limits_{k=0}^\infty \frac{\partial}{\partial x }(\frac{U_{k+1}(x)}{x})\frac{t^{k}}{{k}!}$
$U_n(x)=2\frac{\partial}{\partial x }(\frac{U_{n+1}(x)}{x}) \tag 1$
$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$
$\frac{\partial}{\partial x }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial x }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$
$\frac{\partial}{\partial x }(\sqrt{1+t}e^{x\sqrt{1+t}})=\frac{\partial}{\partial x }(\sum \limits_{k=0}^\infty \frac{\partial (U_k(x))}{\partial x }\frac{t^k}{k!})$
$(1+t)e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{\partial^2 (U_k(x))}{\partial x^2 }\frac{t^k}{k!}$
$\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}+\sum \limits_{k=0}^\infty \frac{U_k(x)t^{k+1}}{k!}=\sum \limits_{k=0}^\infty \frac{\partial^2 (U_k(x))}{\partial x^2 }\frac{t^k}{k!}$
$U_{n+1}(x)+(n+1)U_{n}(x)=\frac{\partial^2 (U_{n+1}(x))}{\partial x^2 } \tag 2$ If we use the Equation (1) in Equation (2), we can get second order linear differential equation.
$U_{n+1}(x)+2(n+1)\frac{\partial}{\partial x }(\frac{U_{n+1}(x)}{x})=\frac{\partial^2 (U_{n+1}(x))}{\partial x^2 } $
$\frac{\partial^2 (U_{n}(x))}{\partial x^2 }-2n\frac{\partial}{\partial x }(\frac{U_{n}(x)}{x})-U_{n}(x)=0 \tag3$
$\frac{\partial^2 (U_{n}(x))}{\partial x^2 }-2n\frac{\partial}{\partial x }(\frac{U_{n}(x)}{x})-U_{n}(x)=0 \tag4$
$U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0 \tag5$
$x^2U''_{n}(x)-2nxU'_{n}(x)+(2n-x^2)U_{n}(x)=0 \tag6$
It is clear that one one solution is
$U_{n}(x)=\frac{\partial^n (e^{x\sqrt{1+t}})}{\partial t^n }|_{t=0}$
Some examples, $U_{0}(x)=e^{x}$
$U_{1}(x)=\cfrac{xe^{x}}{2}$
Can I express the solution of the differential equation as known functions?
I also want to find orthogonal relation for $U_{n}(x)$ if possible.
Thanks a lot for answers.
EDIT: According to GEdgar's answer, I have used the transform
$U_n(x)=x^{(n+\frac{1}{2})}T_n(x)$
$U'_n(x)=(n+\frac{1}{2})x^{(n-\frac{1}{2})}T_n(x)+x^{(n+\frac{1}{2})}T'_n(x)$
$U''_n(x)=(n^2-\frac{1}{4})x^{(n-\frac{3}{2})}T_n(x)+2(n+\frac{1}{2})x^{(n-\frac{1}{2})}T'_n(x)+x^{(n+\frac{1}{2})}T''_n(x)$
$U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0 $
$(n^2-\frac{1}{4})x^{(n-\frac{3}{2})}T_n(x)+2(n+\frac{1}{2})x^{(n-\frac{1}{2})}T'_n(x)+x^{(n+\frac{1}{2})}T''_n(x)-\frac{2n}{x}((n+\frac{1}{2})x^{(n-\frac{1}{2})}T_n(x)+x^{(n+\frac{1}{2})}T'_n(x))+(\frac{2n}{x^2}-1)x^{(n+\frac{1}{2})}T_n(x)=0 $
$(n^2-\frac{1}{4})x^{(n-\frac{3}{2})}T_n(x)+2(n+\frac{1}{2})x^{(n-\frac{1}{2})}T'_n(x)+x^{(n+\frac{1}{2})}T''_n(x)-\frac{2n}{x}((n+\frac{1}{2})x^{(n-\frac{1}{2})}T_n(x)+x^{(n+\frac{1}{2})}T'_n(x))+(\frac{2n}{x^2}-1)x^{(n+\frac{1}{2})}T_n(x)=0 $
$x^{(n+\frac{1}{2})}T''_n(x) +\Big((2n+1)-2n\Big)x^{(n-\frac{1}{2})}T'_n(x)+\Big((n^2-\frac{1}{4})-2n(n+\frac{1}{2})+2n\Big)x^{(n-\frac{3}{2})}T_n(x)-x^{(n+\frac{1}{2})}T_n(x)=0 $
$x^2T''_n(x) +xT'_n(x)-((n-\frac{1}{2})^2+x^{2})T_n(x)=0 $
Thus after transformation ,really We got Modified Bessel Differential Equation http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html
The solutions are the modified Bessel functions of the first and second kinds, and can be written $T_n(x)=C_1 I_{(n-\frac{1}{2})}( x)+C_2 K_{(n-\frac{1}{2})}( x)$ $I_{n}( x)$ is a modified Bessel function of the first kind, and $K_{n}( x)$is modified Bessel function of the second kind.
$U_n(x)=x^{(n+\frac{1}{2})}T_n(x)$ $U_n(x)=C_1 x^{(n+\frac{1}{2})}I_{(n-\frac{1}{2})}( x)+C_2 x^{(n+\frac{1}{2})}K_{(n-\frac{1}{2})}( x)$ Thanks a lot to GEdgar for answer.
Now I need to find $C_1$ and $C_2$ that $U_{n}(x)=\frac{\partial^n (e^{x\sqrt{1+t}})}{\partial t^n }|_{t=0}=C_1 x^{(n+\frac{1}{2})}I_{(n-\frac{1}{2})}( x)+C_2 x^{(n+\frac{1}{2})}K_{(n-\frac{1}{2})}( x)$