God has given us many powers under our control and we can exercise them using our free will. This verse extols us to take control of our live and take positive action. But some things are under God's domain where we should not venture and allow the divine will to take place. For this reason though these two verses seems to contradict each other they are complimentary if we consider them in right context.
Proverbs 3:5 "Trust in the Lord with all your heart and lean not on your own understanding."
Ezra 10:4 "Get up, for this matter is your responsibility, and we support you. Be strong and take action!"
Do they contradict each other? They seem like they do because one says to take action and one says to stay back and let God handle it.
I like to think of Acts 12. There were 2 doors Peter needed to go through. The first door would get Peter out of captivity. But there was no way he could open it because he was a prisoner. So God arranged to get Peter through the first door.
Next came the door to the place where Peter's friends were praying for him. He had to do some convincing to get them to open up. But it was within Peter's control to cause the people inside to let him through the door. So God left it up to him to make it happen.
What this is basically saying, is followers of God shouldn't be lazy. Everything you do is for God, but that doesn't mean He'll do it all himself. The followers of Christ are the messengers and missionaries of God. Trust that the Lord will have it all under control, and he will provide for you. Don't go with your own instinct, but go into the world with the wisdom of God.