source: ncert
topic: calculus
class_level: class_12
chapter: ch9
difficulty: advanced
Class 12 | Ch9: Differential Equations ORDER: order of highest derivative. DEGREE: power of highest derivative (after clearing fractions/roots). FORMATION: differentiate given equation to eliminate arbitrary constants (number of constants=order). METHODS OF SOLVING: METHOD 1 - Variable Separable: f(x)dx=g(y)dy. Integrate both sides separately. dy/dx=e^(x+y)=e^xe^y → e^(-y)dy=e^x dx → -e^(-y)=e^x+C. METHOD 2 - Homogeneous Equations: dy/dx=f(y/x). Substitute v=y/x, y=vx. dy/dx=v+xdv/dx. Equation becomes separable. Identify homogeneous: if f(kx,ky)=k^nf(x,y) (all terms same degree). METHOD 3 - Linear DE: dy/dx+P(x)y=Q(x). Integrating factor IF=e^(integral P dx). Solution: yIF=integral(QIF dx)+C. For dx/dy+P(y)x=Q(y): IF=e^(integral P dy). xIF=integral(QIF dy)+C. STANDARD FORMS: Variable separable: dy/dx=f(x)g(y) → dy/g(y)=f(x)dx. SOLVED EXAMPLES: Example 1: Solve dy/dx=e^(x+y). e^(-y)dy=e^x dx → -e^(-y)=e^x+C → e^x+e^(-y)=C1. Example 2: Solve (x^2+y^2)dx=2xy dy (homogeneous). dy/dx=(x^2+y^2)/(2xy). Put y=vx: v+xdv/dx=(1+v^2)/(2v). xdv/dx=(1+v^2)/(2v)-v=(1-v^2)/(2v). 2v/(1-v^2)dv=dx/x. -ln|1-v^2|=ln|x|+C1 → ln|x(1-v^2)|=-C1 → x(1-y^2/x^2)=C → x^2-y^2=Cx. Example 3: Solve dy/dx+2y/x=x^2 (linear). P=2/x, Q=x^2. IF=e^(integral 2/x dx)=e^(2ln x)=x^2. yx^2=integral(x^2x^2 dx)=integral(x^4 dx)=x^5/5+C. y=x^3/5+C/x^2. COMMON MISTAKES: Degree: power of HIGHEST ORDER derivative (after removing radicals). Homogeneous: all terms must have SAME total degree. Linear DE: coefficient of dy/dx must be 1 (divide through if needed).