Differential Equations
Differential Equations tests order and degree, formation of equations, and solution methods for variable separable, homogeneous, and linear first-order equations. The exam often asks for a particular solution from an initial condition.
Core rules
Order is the highest derivative present; degree is the power of the highest derivative after the equation is made polynomial in derivatives. A general solution of an nth-order equation contains n arbitrary constants. Variable separable equations are written as f(y)dy=g(x)dx and integrated. A homogeneous equation dy/dx=F(x,y) is solved by y=vx, so dy/dx=v+x dv/dx. A linear equation dy/dx+Py=Q has integrating factor IF=e^{∫P dx}; the solution is y·IF=∫Q·IF dx. Apply the initial condition only after finding the general solution.
Worked example
Solve dy/dx+y/x=x², x>0. This is linear with P=1/x and Q=x². IF=e^{∫(1/x)dx}=e^{log x}=x. Multiplying gives x dy/dx+y=x³, that is d(xy)/dx=x³. Integrate: xy=x⁴/4+C, so y=x³/4+C/x.
Traps
Degree is undefined when the equation is not polynomial in derivatives, for example sin(dy/dx). Do not drop the constant C. In the integrating factor, the sign of P matters. When separating variables, check whether division by a factor can lose a constant solution. For homogeneous equations, substitute back y/x after integrating.