What would be one solution for a positive externality other than government intervention?

A positive externality, in economic terms, is a benefit that is “not included in the market price of a good.”  In other words, this is a benefit that someone gets when a good or service is bought but which they do not pay for.  We want to encourage positive externalities to occur because they make our society better-off.  One example of a positive externality is education.  One way to “solve” it (in other words, to promote or encourage it) without government intervention is to encourage people to donate money to charitable organizations that would provide education.

Education is a positive externality because all of society benefits when people become more educated.  An educated person is less likely to commit crimes and be sent to jail.  They are less likely to need government assistance.  They are more likely to get a good job and otherwise contribute to society.  Therefore, society gets good results when people become educated even if society does not pay for those people to do so.

One way (other than having governments provide schools) to encourage this positive externality would be for people to donate more money to charities.  In the past, when the Catholic Church was much stronger in the United States, there were many Catholic schools across the country, even in poor neighborhoods.  These schools could exist because Catholics gave money to their church (and because Catholic religious essentially volunteered their time as teachers).  Today, foundations such as the Gates Foundation give large sums of money to help promote education.  If we wanted to “solve” the positive externality of education without government intervention, we would need to encourage more organizations to run schools and/or to give money to fund schools.