One....x1 + x2 + x3 + x4 + x5 = 34=> x15=34=>x=34/15

You got 5 solutions of this form (0,0,0,0,34) , (0,0,0,34,0)..
and then you got this form (0,0,0,1,33) that has
33*2*(5 nCr 2)=66*(5!/(2!*3!)=66*(10)=660
and then you got this form (0,0,1,32,1) ...
and so on ..

 So you got a lot of answers :)

Answer count depends on
1. data type you are using.
2. related programming language
3. compiler whether 32 bit or 64 bit
Here goes related program and output I have done using cygwin at Windows.

#include <sys/errno.h>
#include <string.h>
#include <stdlib.h>
#include <stdio.h>
#include <stdint.h>
#include <iostream>
using namespace std;
struct Required_Data_ThirtyFour
{
	char CurrentAnswer[1024];
	struct Required_Data_ThirtyFour *prev;
	struct Required_Data_ThirtyFour *next;
	Required_Data_ThirtyFour();
	~Required_Data_ThirtyFour();
};
struct Required_Data_ThirtyFour *CurAnsNode = NULL;
struct Required_Data_ThirtyFour *HeadNode = NULL;
struct Required_Data_ThirtyFour *TailNode = NULL;
char SearchIfAnswerPresent( char *CurrentAnswer)
{
	if ( NULL == CurrentAnswer)
	{
		cout << "CurrentAnswer is NULL\n";
		exit(0);
	}
	Required_Data_ThirtyFour *CurAnsNode = HeadNode;
	while ( CurAnsNode)
	{
		if ( 0 == strcmp( CurrentAnswer, CurAnsNode->CurrentAnswer))
		{
			return 'y';
		}
		CurAnsNode = CurAnsNode->next;
	}
		return 'n';
}
Required_Data_ThirtyFour::Required_Data_ThirtyFour()
{
	prev=NULL;
	next=NULL;
	CurrentAnswer[0]='\0';
}
Required_Data_ThirtyFour::~Required_Data_ThirtyFour()
{
}
int main()
{
	// In this program unsigned int being four bytes.
	unsigned int x1 = 0;
	unsigned int x2 = 0;
	unsigned int x3 = 0;
	unsigned int x4 = 0;
	unsigned int x5 = 0;
	unsigned int SolutionCount = 0;
	char CurrentAnswer[1024] = {};
	HeadNode = CurAnsNode = new Required_Data_ThirtyFour();
	while ( 35 >= x5)
	{
		CurAnsNode->next = new Required_Data_ThirtyFour();
		if ( NULL == CurAnsNode->next)
		{
			cout << strerror(errno) << "\n";
			exit(1);
		}
		CurAnsNode->next->prev=CurAnsNode;
		TailNode=CurAnsNode=CurAnsNode->next;
		x4 = 0;
		while ( 35 >= x4)
		{
			x3 = 0;
			while ( 35 >= x3)
			{
				x2=0;
				while ( 35 >= x2)
				{
					x1 = 0;
					while ( 35 >= x1)
					{
						if ( 34 == ( x1 + x2 + x3 + x4 + x5))
						{
							sprintf( CurrentAnswer, "%02d+%02d+%02d+%02d+%02d=%02d", x1, x2, x3, x4, x5, x1+x2+x3+x4+x5);
							char CurNodeWasSaved = SearchIfAnswerPresent( CurrentAnswer);
							if ( 'n' == CurNodeWasSaved)
							{
								strcpy( CurAnsNode->CurrentAnswer, CurrentAnswer);
								SolutionCount++;
							}
							else
							{
								break;
							}
							cout << CurAnsNode->CurrentAnswer  << "\n";
						}
						x1++;
					}
					x2++;
				}
				x3++;
			}
			x4++;
		}
		x5++;
	}
	Required_Data_ThirtyFour *CurAnsNode = TailNode;
	if ( CurAnsNode )
	{
		TailNode=TailNode->prev;
		delete CurAnsNode;
		CurAnsNode = TailNode;
	}
	// Output from last 3 lines
	// 00+00+01+00+33=34
	// 00+00+00+01+33=34
	// 00+00+00+00+34=34
	// Programmatically (C++ or C), you can
	// Replace:
	//		00
	// With:
	//		UINT32_MAX+1
	// OR
	// Replace:
	//		01
	// With:
	//		UINT32_MAX+2
	// OR
	// ...
	// Replace:
	//		34
	// With:
	//		UINT32_MAX+35
	// Value for UINT32_MAX : 4294967295
	cout << "Total answer count: " << SolutionCount;
	return 0;
}

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Related command:

$ /usr/bin/g++ -g -Wall Non_Negative_Int_Sum.cpp -o a.out
$ ./a.out > Non_Negative_Int_Sum.txt

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Attaching related Output.

The syntax then its wrong.It suppose to be x+y+z+v+w=34

Non_Negative_Int_Sum.txt
Total answer count: 73815
Example:
Count1:            34+00+00+00+00=34
Count51993:   02+03+14+06+09=34
Count73815:   00+00+00+00+34=34

JS code

var x,y,z,v,w;
var count=0;
for(var x=0;x<=34;x++){
    for(var y=0;y<=34;y++){
       for(var z=0;z<=34;z++){
          for(var v=0;v<=34;v++){
            for(var w=0;w<=34;w++){
               if(x+y+z+v+w===34){
                 count++;
               } 
            }
            
          }
       }
    }
}
console.log(count);

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Result 73815 solutions

Related update to the comment from Leonidas Dosas.

<HTML>
	<HEAD>
		<TITLE>
			Sum solution for 34 using nonnegative integers
		</TITLE>
	</HEAD>
	<BODY>
		<TABLE border="1" height="100%" width="5%">
			<TR>
				<TD valign=top align=left>
					x1
				</TD>
				<TD valign=top align=left>
					x2
				</TD>
				<TD valign=top align=left>
					x3
				</TD>
				<TD valign=top align=left>
					x4
				</TD>
				<TD valign=top align=left>
					x5
				</TD>
				<TD valign=top align=left>
					Sum
				</TD>
				<TD valign=top align=left>
					Answer count
				</TD>
			</TR>
			<script language="javascript">
				var x=0,y=0,z=0,v=0,w=0;
				var AnswerCount=0;
				for(var x=0;x<=34;x++)
				{
					for(var y=0;y<=34;y++)
					{
						for(var z=0;z<=34;z++)
						{
							for(var v=0;v<=34;v++)
							{
								for(var w=0;w<=34;w++)
								{
									if(x+y+z+v+w===34)
									{
										document.write("<TR>");
										document.write("<TD valign=top align=left>");
										CurVar=""+x;
										if ( 1 == CurVar.length)
										{
											document.write("0"+x);
										}
										else
										{
											document.write(x);
										}
										document.write("</TD><TD valign=top align=left>");
										if ( 1 == (""+y).length)
										{
											document.write("0"+y);
										}
										else
										{
											document.write(y);
										}
										document.write("</TD><TD valign=top align=left>");
										if ( 1 == (""+z).length)
										{
											document.write("0"+z);
										}
										else
										{
											document.write(z);
										}
										document.write("</TD><TD valign=top align=left>");
										if ( 1 == (""+v).length)
										{
											document.write("0"+v);
										}
										else
										{
											document.write(v);
										}
										document.write("</TD><TD valign=top align=left>");
										if ( 1 == (""+w).length)
										{
											document.write("0"+w);
										}
										else
										{
											document.write(w);
										}
										AnswerCount++;
										document.write("</TD><TD valign=top align=left>=34</TD><TD align=left valign=top>"+AnswerCount+"</TD>");
										document.write("</TR>");
									}
								}
							}
						}
					}
			}
			document.write("<TR>");
			document.write("<TD valign=top align=left colspan=7>");
			document.write("Total answer count: "+AnswerCount);
			document.write("</TD>");
			document.write("</TR>");
			</script>
		</TABLE>
	</BODY>
</HTML>

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Hence informing the following:
Answer count depends on
1. data type you are using.
2. related programming language
3. compiler whether 32 bit or 64 bit
4. related applications(either a.out or program.exe or java or browser or ...)

This is a math problem how is your pc specs related!?