Differential Equations
Differential Equations tests order and degree, formation of differential equations, and solution of first-order first-degree equations. The exam usually names the type or gives a form that reveals it.
Classification and methods
Order is the highest derivative present; degree is the power of the highest derivative after the equation is made polynomial in derivatives. A variable-separable equation can be written f(x)dx=g(y)dy. A homogeneous equation is solved by y=vx, so dy/dx=v+x dv/dx. A linear equation dy/dx+Py=Q has integrating factor e^∫P dx, and the solution is y·IF=∫Q·IF dx. An exact equation M dx+N dy=0 satisfies ∂M/∂y=∂N/∂x. To form a differential equation, eliminate arbitrary constants by differentiating as many times as there are constants.
Worked example
Solve dy/dx+y/x=x. This is linear with P=1/x and Q=x. IF=e^∫(1/x)dx=e^{ln x}=x. Then d(xy)/dx=x·x=x², so xy=x³/3+C, or y=x²/3+C/x.
Traps
Degree is not defined until fractional powers of derivatives are cleared. Dividing by a variable expression can lose a singular solution. In the linear formula, the equation must first be in the standard form with coefficient 1 on dy/dx. The integrating factor multiplies both sides; forgetting Q·IF is a frequent error.