Now we rearrange to get {{MATH_403}}. This shows us that any solution to our differential equation {{CMD_404}}must{{CMD_405}} be one of the functions we've already found. Our argument relies on some features that are special to this problem; proving that we've found all the possible solutions to an arbitrary differential equation is a very difficult task! {{CMD_406}} {{CMD_407}} {{CMD_408}} A function {{MATH_409}} is called a {{CMD_410}} of the differential equation. Since there are infinitely many solutions to a differential equation, we can impose an additional condition (say {{MATH_411}}), called an {{CMD_412}}. When we are asked to find a function {{MATH_413}} that satisfies both the differential equation (DE) and the initial condition (IC), this is called an {{CMD_414}} (IVP). Let's try one out. {{CMD_415}} Solve the initial value problem (IVP): {{ENV_416}} {{CMD_417}} The figure below shows several solutions of the differential equation. {{CMD_418}} {{ENV_419}} {{CMD_420}} The figure suggests that the only solution to the initial value problem is the function {{MATH_421}}. We can verify this result.