| --- |
| source: ncert |
| topic: calculus |
| class_level: class_12 |
| chapter: ch9 |
| difficulty: advanced |
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| |
| 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^x*e^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+x*dv/dx. Equation becomes separable. |
| Identify homogeneous: if f(kx,ky)=k^n*f(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: y*IF=integral(Q*IF dx)+C. |
| For dx/dy+P(y)*x=Q(y): IF=e^(integral P dy). x*IF=integral(Q*IF 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+x*dv/dx=(1+v^2)/(2v). |
| x*dv/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. |
| y*x^2=integral(x^2*x^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). |
