How to write equation as shown in picture using Latex?

Here you go:

documentclass[10pt,a4paper]{article}
usepackage{amsmath}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{graphicx}
begin{document}
    $ufrac{partial u}{partial x}+vfrac{partial u}{partial y}+wfrac{partial u}{partial z}+lambda_1begin{pmatrix}
    u^2frac{partial^2 u}{partial x^2}+vfrac{partial^2 u}{partial y^2}+w^2frac{partial^2 u}{partial z^2}+2uvfrac{partial^2 u}{partial xpartial y}+2vwfrac{partial^2 u}{partial ypartial z}+2uwfrac{partial^2 u}{partial xpartial z}
    end{pmatrix} = vfrac{partial^2 u}{partial z^2},$
end{document}

this will give you:

PS: Welcome to TeX.se, from next time, please provide a MWE.

Here is a start:

documentclass[]{article}
usepackage{amsmath}

begin{document}
begin{equation}
    ufrac{partial u}{partial x}+...
    +lambda_1 
    begin{pmatrix}
    u^2 frac{partial^2 u}{partial x^2}+...
    + 2 uv frac{partial^2 u}{partial x partial y}+...
    end{pmatrix} = v frac{partial^2 u}{partial z^2}
end{equation}
end{document}

Borrow a little from all answer and make a little change (such as change v to nu in the right-hand side)

documentclass{article}
usepackage{amsmath}
begin{document}
begin{equation}
    ufrac{partial u}{partial x}+vfrac{partial u}{partial y}+wfrac{partial u}{partial z}+lambda_1begin{pmatrix}
    u^2frac{partial^2 u}{partial x^2}+vfrac{partial^2 u}{partial y^2}+w^2frac{partial^2 u}{partial z^2}+2uvfrac{partial^2 u}{partial xpartial y}+2vwfrac{partial^2 u}{partial ypartial z}+2uwfrac{partial^2 u}{partial xpartial z}
    end{pmatrix} = nufrac{partial^2 u}{partial z^2},
    tag{1.25}
end{equation}
end{document}

A variation on the theme, but with a greater emphasis on simplifying the input:

documentclass{article}
usepackage{amsmath}
usepackage{xparse}

newcommand{pdiff}{mathop{}!partial}

ExplSyntaxOn
NewDocumentCommand{pder}{omm}
 {
  frac{pdiffIfValueT{#1}{^{#1}}#2}{faisal_pder_vars:n { #3 }}
 }
cs_new_protected:Nn faisal_pder_vars:n
 {
  clist_map_inline:nn { #1 } { pdiff##1 }
 }
ExplSyntaxOff


begin{document}

begin{equation}
upder{u}{x}+vpder{u}{y}+wpder{u}{z}+
  lambda_1
  left(begin{gathered}
    u^2pder[2]{u}{x^2}+vpder[2]{u}{y^2}+w^2pder[2]{u}{z^2}
    mspace{-medmuskip}{}
    +2uvpder[2]{u}{x,y}+2vwpder[2]{u}{y,z}+2uwpder[2]{u}{x,z}
    end{gathered}right)
= nupder[2]{u}{z^2},
end{equation}

begin{equation}
begin{split}
upder{u}{x}&+vpder{u}{y}+wpder{u}{z}
+lambda_1biggl(
    u^2pder[2]{u}{x^2}+vpder[2]{u}{y^2}+w^2pder[2]{u}{z^2}

&+2uvpder[2]{u}{x,y}+2vwpder[2]{u}{y,z}+2uwpder[2]{u}{x,z}biggr)
= nupder[2]{u}{z^2},
end{split}
end{equation}

end{document}

The syntax for pder is

1. optional argument for the order of the derivative (if greater than 1)
2. function to differentiate
3. list of the variables the derivative is taken with respect to, comma separated

In the first case I used mspace{-medmuskip}{}+ in order to have the plus sign correctly spaced. I used gathered instead of pmatrix because it is semantically sounder.

With a few shortcut macros it's much easier:

plain TeX version:

let~catcode~`86~`j0~`X13~`C1~`D2~`M3jdefX81C~`8113jdefDXZZ81C~
`8113DXYY81CZ81jletDYLLjletYFFjfiYNNjdefYPPjpartialZIZAZBZj{Zj}Y
EEjexpandafterYOOjelseYKKjifxNI818283CL}81NAC82DNBC83Djfuturelet
jTjHDNjHCKjT}L{AOL{BF{DXUUCI)jpCjp)CDDDY VVjoverNjp)818283CKX81X
CP82VP83DOCP^C81D82VjbC81DC83DDFDYRRjrelaxNjb8182Cjd8182RDNjd81%
8283CK83RP82^C81DOP82P83FDY!!uY@@vY##wY$$xYj%%yY&&zXvv81C81^2DMM
!U!$+@U!%+#U!&+jlambda_1jleft(v!U)2! $+v@U)2!%+v#U)2!&jatop+2!@U
)2!C$%D+2@#U)2!C%&D+2!#U)2!C$&Djright)=jnuU)2!&jeqno(1.25)MMjbye

LaTeX version:

documentclass{article}begin{document}
let~catcode~`86~`j0~`X13~`C1~`D2~`M3jdefX81C~`8113jdefDXZZ81C~
`8113DXYY81CZ81jletDYLLjletYFFjfiYNNjdefYPPjpartialZIZAZBZj{Zj}Y
EEjexpandafterYOOjelseYKKjifxNI818283CL}81NAC82DNBC83Djfuturelet
jTjHDNjHCKjT}L{AOL{BF{DXUUCI)jpCjp)CDDDY VVjoverNjp)818283CKX81X
CP82VP83DOCP^C81D82VjbC81DC83DDFDYRRjrelaxNjb8182Cjd8182RDNjd81%
8283CK83RP82^C81DOP82P83FDY!!uY@@vY##wY$$xYj%%yY&&zXvv81C81^2DMM
!U!$+@U!%+#U!&+jlambda_1jleft(v!U)2!$+v@U)2!%+v#U)2!&jatop+2!@U)
2!C$%D+2@#U)2!C%&D+2!#U)2!C$&Djright)=jnuU)2!&jeqno(1.25)MMjstop

output:

---

Inspired by David Carlisle's xii :)

$$u frac{partial u}{partial x}+v frac{partial u}{partial y}+w frac{partial u}{partial z}+lambda_{1}left(begin{array}{c}
u^{2} frac{partial^{2} u}{partial x^{2}}+v^{2} frac{partial^{2} u}{partial y^{2}}+w^{2} frac{partial^{2} u}{partial z^{2}} 
+2 u v frac{partial^{2} u}{partial x partial y}+2 v w frac{partial^{2} u}{partial partial partial z}+2 u w frac{partial^{2} u}{partial x partial z}
end{array}right)=nu frac{partial^{2} u}{partial z^{2}}$$