Yes, you can have different DCs for active and passive DCs.
I can not find anything in the DMG or PHB that explicitly says that the DC must be the same for passive and active checks.
It is somewhat implied in a number of places in the DMG where it says thing similar to this:
DMG pg 120-121
A trap's description specifies the checks and DCs needed to detect it, disable it, or both. A character actively looking for a trap can attempt a wisom (Perception) check against the trap's DC. You can also compare teh DC to detect the trap with each character's passive Wisdom (Perception) score to determine whether anyone in the party notices the trap in passing.
The language implies a single DC for either check, but does not outright state it. The list of traps on the next few pages also have a single DC for active and passive checks, but do not outright state that you must make them the same.
However, in the section of the DMG on determining DCs, we see the following:
DMG pg 238
It's your job to establish the Difficulty Class for an ability check or saving throw when a rule or an adventure doesn't give you one. Sometimes you'll even want to change such established DCs. When you do so, think of how difficult a task is and then pick the associated DC from the Typical DCs table.
This directly gives you the authority to set or change DCs. It is perfectly reasonable (and adds to verisimilitude) for a passive check to have a higher DC than an active one.
Additionally, DMG pg 104 says:
A concealed door is a normal door that is hidden from view. A secret door is carefully crafted to blend into its surrounding surface, whereas a concealed door is most often hidden by mundane means. It might be covered by a tapestry, covered with plaster, or (in the case of a concealed trapdoor) hidden under a rug. Normally, no ability check is required to find a concealed door. A character need only look in the right place or take the right steps to reveal the door. However, you can use the characters' passive Wisdom (Perception) scores to determine whether any of them notices tracks or signs of a tapestry or rug having been recently disturbed.
This is a situation where the active DC is 0, but the passive DC is >0.
Adding this to the fact that in the Lost Mine of Phandelver pg 20, it says:
Spotting a secret door from a distance of no more than 10 feet without actively searching for it requires a passive Wisdom (Perception) score of 15 or higher, whereas a character who takes the time to search the wall can find the secret door with a successful DC 10 Wisdom (Perception) check.
And we have a second source that says that we can set different active and passive DCs.
So yes, you can set different DCs for active and passive checks.
Something not stated in the DMG, but considered common practice, is to reduce the DC for more explicit explanations of a goal. As an example, compare this:
Player: I check the room for anything of value
DM: Roll Investigation [DC 12 for small treasure, DC 16 for secret compartment]
DM: you find 4 GP worth of small coins in various drawers of the desk in the corner, and a finely crafted, ornamental dagger worth about 20 GP over the mantle of the Duke's bed.
Player: I check the desk for anything of value, especially looking for secret compartments
DM: Roll Investigation [DC 5 for small treasure, DC 8 for secret compartment]
DM: you find 4 GP worth of small coins in various drawers of the desk in the corner, and little else worth carrying out to sell.
DM: However, you also find a false bottom on one of the drawers. After opening it up, you find documents proving that the duke is the head of the assassination attempt, as well as a tiny opal worth about 20 GP.
The more precise your players are at explaining what they are trying to do, the more likely they are to find what they are looking for (assuming it exists), but the less likely they are to find things outside that explanation that might be more obvious.