Differential Equations
Aisi equation jisme function ke saath uske derivatives bhi hon, use differential equation (DE) kehte hain. Ye real duniya me change ko model karti hain — population growth, cooling, motion, etc. Is chapter me DE ko solve karna seekhenge.
1. Order aur Degree
- Order: equation me sabse highest derivative ka order.
- Degree: highest order derivative ki power (jab equation polynomial form me ho derivatives me).
\(\left(\dfrac{d^{2}y}{dx^{2}}\right)^{3} + \dfrac{dy}{dx} + y = 0\) — highest derivative \(\frac{d^{2}y}{dx^{2}}\) (order \(= 2\)), uski power \(3\) (degree \(= 3\)).
2. General aur Particular Solution
General solution me arbitrary constants hote hain (order jitne). Jab di gayi conditions se wo constants nikaal lete hain, to milta hai particular solution.
3. Method 1: Variable Separable
Agar DE ko \(\dfrac{dy}{dx} = f(x)\,g(y)\) form me likha ja sake, to variables alag karke dono taraf integrate karo: \[ \int \frac{dy}{g(y)} = \int f(x)\,dx + C. \]
\(\dfrac{dy}{dx} = xy\) solve karo. Variables alag karo: \[ \int \frac{dy}{y} = \int x\,dx \;\Rightarrow\; \ln|y| = \frac{x^{2}}{2} + C. \]
4. Method 2: Homogeneous Equations
Agar \(\dfrac{dy}{dx} = F\!\left(\dfrac{y}{x}\right)\) ho, to substitution \(y = vx\) (\(\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\)) use karke variable-separable bana lo.
5. Method 3: Linear Differential Equation
Form: \(\dfrac{dy}{dx} + P(x)\,y = Q(x)\). Iske liye integrating factor: \[ \text{I.F.} = e^{\int P\,dx}. \] Solution: \[ y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C. \]
\(\dfrac{dy}{dx} + y = e^{x}\). Yahan \(P = 1\), so I.F. \(= e^{\int 1\,dx} = e^{x}\). \[ y\,e^{x} = \int e^{x}\cdot e^{x}\,dx = \int e^{2x}\,dx = \frac{e^{2x}}{2} + C, \quad \Rightarrow\; y = \frac{e^{x}}{2} + C e^{-x}. \]
Key Takeaways
- Order = highest derivative ka order; degree = uski power.
- General solution me constants, particular me unhe nikaal lete hain.
- Teen methods: variable separable, homogeneous (\(y = vx\)), linear (I.F. \(= e^{\int P\,dx}\)).
- Linear DE: \(y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + C\).