Differential Equations
Differential Equations asks you to classify an equation by order and degree, form one by eliminating arbitrary constants, and solve first-order equations. The three solution families in the syllabus are variable separable, homogeneous (substitute y = vx), and linear dy/dx + Py = Q solved with an integrating factor.
Definitions and methods
- Order is the highest derivative present. Degree is the power of the highest-order derivative after the equation is made polynomial in derivatives.
- Variable separable: rearrange to f(y)dy = g(x)dx and integrate both sides.
- Homogeneous: if dy/dx = F(x, y) with F homogeneous of degree 0, put y = vx so dy/dx = v + x dv/dx, then separate.
- Linear: dy/dx + Py = Q with P, Q functions of x. Integrating factor IF = e^∫P dx, and the solution is y·IF = ∫Q·IF dx + C.
- Formation: differentiate the given family as many times as there are arbitrary constants, then eliminate them.
Worked example
Solve dy/dx + y/x = x² for x > 0. Here P = 1/x and Q = x², so IF = e^∫(1/x)dx = e^(log x) = x. Multiplying through by x gives x dy/dx + y = x³, that is d/dx(xy) = x³. Integrating, xy = x⁴/4 + C, so y = x³/4 + C/x.
Traps
- Assigning a degree to an equation containing sin(dy/dx) or e^(dy/dx); degree is undefined there.
- Confusing order with degree when the highest derivative appears squared.
- Dropping the constant of integration or placing C inside the integrating factor.
- Forgetting that the integrating factor multiplies Q as well as y.