Differential Equations
Differential Equations tests order and degree, formation of equations, variable separable equations, homogeneous equations and linear first-order equations.
Core rules
- Order is the highest derivative; degree is the power of the highest derivative after the equation is polynomial in derivatives.
- A general solution contains as many arbitrary constants as the order; a particular solution uses initial conditions.
- Variable separable: separate y and x, integrate both sides, then apply conditions.
- Linear first-order form dy/dx+Py=Q has integrating factor e^(∫P dx); multiply and integrate d(y times IF)/dx=Q times IF.
Worked example
Solve dy/dx=2xy with y(0)=1. Separate: dy/y=2x dx. Integrate: ln|y|=x^2+C, so y=Ae^(x^2). Using y(0)=1 gives A=1. The particular solution is y=e^(x^2). Check: dy/dx=2x e^(x^2)=2xy.
Traps
- Do not lose the constant or the absolute value when integrating 1/y.
- Degree is not defined when the equation is not polynomial in derivatives.
- In the linear formula, P and Q must be functions of x only after dividing by the coefficient of dy/dx.
- A homogeneous equation needs the substitution y=vx or x=vy, not simple separation.
- Check whether a constant solution such as y=0 was divided away.