Differential Equations - BSEB Inter Study Guide
Differential Equations
This domain tests order and degree, formation of differential equations, variable-separable equations, homogeneous equations and linear first-order equations.
Core rules
- 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: separate to f(y)dy=g(x)dx and integrate both sides.
- Homogeneous equation dy/dx=F(x,y): put y=vx, so dy/dx=v+x dv/dx.
- Linear equation dy/dx+Py=Q has integrating factor IF=e^(∫P dx); solution is y·IF=∫Q·IF dx.
- A general solution contains as many arbitrary constants as the order.
Worked example
Solve dy/dx+y/x=x². This is linear with P=1/x, Q=x². IF=e^(∫(1/x)dx)=e^(ln x)=x. Then y·x=∫x·x² dx=∫x³ dx=x⁴/4+C. So y=x³/4+C/x.
Traps
- Calling dy/dx+sin(dy/dx)=0 degree 1: degree is not defined when the equation is not polynomial in derivatives.
- Forgetting the constant of integration.
- Using IF=e^(∫P dx) but dropping the sign of P.
- In homogeneous equations, forgetting to replace dy/dx by v+x dv/dx.
- Applying an initial condition before finding the general solution.