Hello,
I have a problem with a VF page. I improved the vf page with the inline editing but I'm not able to save if I edit a record. In a first moment it seems it's saving but if I click the search button the page reload the record without change. I think the problem is in the save method. Can someone hepl me?
Thanks

vf page
 

<apex:page controller="SearchWithWrapperC" lightningStylesheets="true"  docType="html-5.0" id="pg">
        <script type="text/javascript">
        function selectAllCheckboxes(obj,receivedInputID){
            var inputCheckBox = document.getElementsByTagName("input");                  
            for(var i=0; i<inputCheckBox.length; i++){          
                if(inputCheckBox[i].id.indexOf(receivedInputID)!=-1){                                     
                    inputCheckBox[i].checked = obj.checked;
                }
            }
        }
    </script>
    <apex:form >
        <apex:pageBlock title="Billing Details Search Page" >
            <br/>
            <br/>
            <apex:actionRegion >
                <center>
                    <apex:outputLabel value="Start Date"/>
                    <apex:input value="{!StartDate}" type="date" /><br/>
                    <apex:outputLabel value="End Date "/>
                    <apex:input value="{!EndDate}" type="date" /> 
                </center>
            </apex:actionRegion>  
            <br/>
            
            <div align="center" draggable="false" >            
                <apex:commandButton value="Search" action="{!loadData}" style="text-align:left;" />               
                <apex:commandButton value="Moves to next month" style="text-align:left;" onclick="if(!confirm('If you click ok all the selected billing details will be paid for the current month. Are you sure? ')) return false;" action="{!processSelected}" />  
            </div>        
        </apex:pageBlock>
        
        <apex:pageBlock mode="inlineEdit" id="pgblk" >
            <apex:pageBlockButtons >
<apex:commandbutton value="save" action="{!save1}"/>
</apex:pageBlockButtons>
            <apex:pageBlockTable value="{!listBD}" var="b">
                <apex:column >
                    <apex:facet name="header">
                        <apex:inputCheckbox onclick="selectAllCheckboxes(this,'inputId')"  />
                    </apex:facet>
                    <apex:inputCheckbox value="{!b.check_box}" id="inputId" />
                </apex:column>
                
				<apex:column Headervalue="SF OPPORTUNITY ID">
						<apex:outputfield value="{!b.cs.SF_Opportunity_Id__c}"/>
				</apex:column>
				<apex:column headervalue="BILLING TYPE">
						<apex:outputfield value="{!b.cs.Billing_Type__c}"/>
				</apex:column>
				<apex:column headervalue="BILLING PERIOD" >
						<apex:outputfield value="{!b.cs.Billing_Period__c}"/>
				</apex:column>
				<apex:column headervalue="MONTHLY FORECAST" >
						<apex:outputfield value="{!b.cs.Monthly_Forecast__c}"/>
				</apex:column>
				<apex:column headervalue="END OF BILLING" >
						<apex:outputfield value="{!b.cs.End_of_Billing__c}"/>
				</apex:column>
 				<apex:column headervalue="AMOUNT" >
						<apex:outputfield value="{!b.cs.Amount__c}"/>
				</apex:column>
				<apex:column headervalue="BILLING STATUS" >
						<apex:outputfield value="{!b.cs.Billing_Status__c}"/>
				</apex:column>    
            </apex:pageBlockTable>   
            
        </apex:pageBlock>        
    </apex:form>
</apex:page>

Controller

public class SearchWithWrapperC {
    
    public  Date StartDate {get;set;}
    public  Date EndDate {get;set;}
   
    
    public List<WrapperClass> listBD {get;set;}
  
    public List<Billing_Detail__c> blst=new List<Billing_Detail__c>();

	public List<Billing_Detail__c> getBill(){
 	blst=[Select Id,SF_Opportunity_Id__c, Name,Billing_Type__c, Billing_Period__c , Monthly_Forecast__c,End_of_Billing__c, Amount__c, Billing_Status__c   from Billing_Detail__c Where Billing_Status__c='Authorized for Billing'AND Monthly_Forecast__c>=:StartDate AND Monthly_Forecast__c <= :EndDate  Order by Monthly_Forecast__c] ;
      return blst;

	}
//save method inline editing
	public void save1() {
     update blst;
	}	
    
    
   

    
    public void loadData() {
        listBD = new List<WrapperClass>();
        for(Billing_Detail__c  cr : [Select Id,SF_Opportunity_Id__c, Name,Billing_Type__c, Billing_Period__c , Monthly_Forecast__c,End_of_Billing__c, Amount__c, Billing_Status__c   from Billing_Detail__c Where Billing_Status__c='Authorized for Billing'AND Monthly_Forecast__c>=:StartDate AND Monthly_Forecast__c <= :EndDate  Order by Monthly_Forecast__c ]){
            listBD.add(new WrapperClass (cr, false));
        }
    }
    
    	public List<WrapperClass> getBilling() {
            if(listBD == null) {

		listBD = new List<WrapperClass>(); 
        for(Billing_Detail__c  cr : [Select Id,SF_Opportunity_Id__c, Name,Billing_Type__c, Billing_Period__c , Monthly_Forecast__c,End_of_Billing__c, Amount__c, Billing_Status__c   from Billing_Detail__c Where Billing_Status__c='Authorized for Billing'AND Monthly_Forecast__c>=:StartDate AND Monthly_Forecast__c <= :EndDate  Order by Monthly_Forecast__c ]){
            listBD.add(new WrapperClass (cr, false));
        }
        }		
   		 return listBD;
		}
	
	    
   public PageReference processSelected() {
       Integer m = Date.Today().Month();
       Integer y = Date.Today().Year();
       
       
       List<Billing_Detail__c> selectedBD = new List<Billing_Detail__c>();
       for(WrapperClass bd: getBilling()){
                    
          Billing_Detail__c billlingDetail ;
           if(bd.check_box== true){
              
            billlingDetail = new Billing_Detail__c(Id=bd.cs.Id);
          billlingDetail.Monthly_Forecast__c = Date.newinstance(  y , m ,  bd.cs.Monthly_Forecast__c.day() );
               
          
               selectedBD.add(billlingDetail);
                 
           } 
       }
       
         
          update selectedBD;
           loadData();
       
       
     
       return null;
   }
    

    //wrapper
    public class WrapperClass {
        public Billing_Detail__c  cs {get; set;}
        public Boolean check_box {get; set;}
        
        public WrapperClass(Billing_Detail__c  c, Boolean check_box){
            this.cs = c;
            this.check_box = false;
        }
    }
}