Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — separating the variables
Solve dxdy=xy.
- 1The right side factorises as a function of x times a function of y, so the variables separate.
- 2y1dy=xdxAll the y's to the left, all the x's to the right.
- 3Integrate both sides: ln∣y∣=2x2+COne constant is enough for both sides.
- 4y=ex2/2+C=eCex2/2Split the exponential.
- 5y=Aex2/2Absorb eC into a single constant A straight away.
Example 2 — the integrating factor
Solve dxdy+2y=ex.
- 1The equation is linear and already has coefficient 1 on dxdy, with P=2.Divide through first if it were not.
- 2IF=e∫2dx=e2xNo constant needed in the integrating factor.
- 3Multiplying makes the left side exact: dxd(e2xy)=e2xex=e3xThis is the whole point of the method.
- 4Integrate: e2xy=3e3x+CNow the constant appears.
- 5y=3ex+Ce−2xDivide through by e2x.
- 6Check: y′+2y=ex ✓The C terms cancel, as they must.
Example 3 — a second-order equation
Solve dx2d2y−5dxdy+6y=12.
- 1Complementary function first: solve the equation with 0 on the right.
- 2Auxiliary equation m2−5m+6=0, so (m−2)(m−3)=0.Distinct real roots.
- 3CF =Ae2x+Be3xTwo arbitrary constants, as a second-order equation requires.
- 4Particular integral: the right side is a constant, so try y=k.Then y′=y′′=0.
- 56k=12, so k=2.
- 6y=Ae2x+Be3x+2General solution = complementary function + particular integral.