Differential Equations
1. Real Life Application - Where Differential Equations Live? (Impactful Intro)
Differential equation is equation with derivative dy/dx, it describes how something changes. Real life is full of change so DE everywhere: Population growth bacteria - rate of growth dy/dt is proportional to current population y, so dy/dt = k y, solution exponential growth. Radioactive decay - rate of decay proportional to amount left. Newton's law of cooling - rate of cooling proportional to temperature difference. Electric circuits - current change described by L di/dt + R i = V. Medicine drug absorption - concentration change. Economics investment growth. Physics motion F=m a = m d²x/dt². So DE models any changing system. That is why we study them.
Real life: Population growth dy/dt = k y exponential, Radioactive decay dN/dt = -λ N, Newton's cooling dT/dt = k(Ts-T), Electric circuit L di/dt + R i = V, Medicine drug concentration change, Economics growth, Physics F=m d²x/dt² motion. DE = equation with derivative describes how something changes.
• Population: dy/dt = k y → y = y0 e^{kt} exponential growth
• Radioactive Decay: dN/dt = -λ N → N=N0 e^{-λt}
• Newton's Cooling: dT/dt = k(Ts - T) temperature drops
• Electric Circuit: L di/dt + R i = V current change
• Medicine: Drug concentration dC/dt change absorption
• Physics: Motion F=m a = m d²x/dt² second order
• KEY: DE = equation with derivative describes rate of change → models any changing system
2. Definition, Order and Degree - Basics
Differential equation is equation that relates function to its derivatives. Order is highest derivative present in equation. Example dy/dx + y = x order 1 because highest derivative dy/dx first order. d²y/dx² + 3 dy/dx + 2y =0 order 2 because highest d²y/dx² second derivative. Degree is power of highest order derivative when equation is polynomial in derivatives. Example (d²y/dx²)² + dy/dx = sin x order 2 degree 2 because highest derivative squared. If derivative inside root or inside sin, degree not defined. Must make polynomial first.
DE is equation relating function to derivatives. Order = highest derivative present. Degree = power of highest order derivative when polynomial in derivatives. Example dy/dx+y=x Order1 Degree1, d²y/dx²+3 dy/dx+2y=0 Order2 Degree1, (d²y/dx²)²+dy/dx=sin x Order2 Degree2. If derivative inside root/trig degree not defined.
• Definition: Equation relating y to derivatives dy/dx, d²y/dx² etc
• Order = highest derivative present
• Example dy/dx+y=x → Order1, d²y/dx²+3dy/dx+2y=0 → Order2
• Degree = exponent of highest order derivative when polynomial in derivatives
• (d²y/dx²)²+dy/dx=sin x → Order2 Degree2 squared
• If derivative inside √, fraction, sin/cos degree not defined
• Make polynomial first then find degree
3. General and Particular Solutions - Family of Curves
General solution contains arbitrary constants equal to order. Example y = A e^x + B e^{-x} contains 2 constants A,B, it is general solution of second order DE d²y/dx² - y =0. Particular solution is general solution with constants given specific values using initial conditions. Example y = C e^x general, if condition y(0)=2 then 2=C e^0 C=2 particular solution y=2 e^x. Geometrically general solution is family of curves (infinite curves shifted), particular is one curve passing through given point. Order 1 → 1 constant, order 2 → 2 constants.
General solution contains arbitrary constants = order, family of curves. Particular solution constants fixed using conditions y(x0)=y0. Order1→1 constant, Order2→2 constants. Example y=C e^x general, y(0)=2 → C=2 → y=2 e^x particular one curve.
• General Solution: Contains arbitrary constants = order number, infinite family of curves
• Example d²y/dx² - y=0 → y=A e^x + B e^{-x} 2 constants general
• Particular Solution: Constants fixed using initial conditions y(x0)=y0
• Example y=C e^x general, y(0)=2 → C=2 → y=2 e^x particular
• Geometrically general = family, particular = one curve through point
• Order1→1 constant, Order2→2 constants
4. Formation of Differential Equations - Eliminating Constants
Formation means given family of curves with arbitrary constants, find DE by eliminating constants using differentiation. Steps: Given equation with n arbitrary constants, differentiate n times, eliminate constants. Example family y = A x + A²? Actually simple y = A x? Let's take y = A e^x? Example y = C x? Example y = A x²? Suppose y = A x² contains 1 constant, differentiate dy/dx =2A x, eliminate A = y/x²? Actually y=A x² → A=y/x² put into dy/dx =2 (y/x²) x =2y/x so DE x dy/dx -2y=0. For 2 constants like y= A sin x + B cos x differentiate twice get d²y/dx² + y =0. Number of constants = order of DE. So count constants to know order.
Formation: Given family with n constants differentiate n times eliminate constants get DE. Steps: Count constants n → differentiate n times → eliminate. Example y=A x² A=y/x² dy/dx=2y/x → x dy/dx-2y=0 Order1. y=A sin x + B cos x diff twice → d²y/dx² + y=0 Order2. Constants number = order.
• Given family with n arbitrary constants → DE of order n
• Steps: 1) Count constants n 2) Differentiate n times 3) Eliminate constants
• Example y=A x²: dy/dx=2A x, A=y/x² → x dy/dx-2y=0
• Example y=A sin x + B cos x: d²y/dx² + y=0 Order2
• Circle x²+y²=a²: x + y dy/dx=0
• Constants number = order of DE
5. Variable Separable Method - Easiest Method
Variable separable: Bring all y terms with dy on one side, x terms with dx other side, then integrate both sides. Form dy/dx = g(x)·h(y) → dy/h(y) = g(x) dx → ∫ dy/h(y)=∫ g(x)dx. Example dy/dx = x·y → dy/y = x dx → ∫ dy/y = ∫ x dx → ln|y| = x²/2 + C → y = C1 e^{x²/2}. Example dy/dx = e^x + e^{-y}? Actually dy/dx = (1+y²)/... Need to separate. Always after separating integrate both sides add C only one side enough. This method works when you can factor x and y separately.
Variable separable: dy/dx = g(x)·h(y) → dy/h(y)=g(x)dx → ∫dy/h(y)=∫g(x)dx. Example dy/dx=x·y → dy/y=x dx → ln|y|=x²/2+C → y=C1 e^{x²/2}. Separate x and y then integrate. Add C once.
• Form: dy/dx = g(x)·h(y) can separate x and y
• Method: dy/h(y)=g(x)dx → ∫ dy/h(y)=∫ g(x)dx
• Example dy/dx=x·y → dy/y=x dx → ln|y|=x²/2+C → y=C1 e^{x²/2}
• Example dy/dx=e^{x+y}=e^x·e^y → e^{-y}dy=e^x dx
• Integrate both sides, add C only once
• Works when x and y factor separately
6. Homogeneous Differential Equations - y=vx Substitution
Homogeneous function is function where degree of each term same in x,y. Example x²+y², x²+xy. Homogeneous DE is of form dy/dx = f(x,y)/g(x,y) where f,g homogeneous same degree. Method: Put y = v x → dy/dx = v + x dv/dx, substitute, equation becomes variable separable in v and x. Steps: Replace y=v x, dy/dx=v + x dv/dx, substitute, separate v and x, integrate, replace v=y/x back. Example dy/dx = (x²+y²)/xy homogeneous, put y=v x → v + x dv/dx = (1+v²)/v → separable. Same can be done for dx/dy if homogeneous in x.
Homogeneous DE: dy/dx = f(x,y)/g(x,y) where f,g homogeneous same degree (degree each term same). Put y=v x → dy/dx=v+x dv/dx substitute → variable separable in v,x integrate → replace v=y/x. Example (x²+y²)/xy = (1+v²)/v after substitution.
• Homogeneous function degree same each term e.g. x²+y², xy, x²+xy
• Homogeneous DE: dy/dx = f(x,y)/g(x,y) same degree homogeneous
• Method: Put y=v x → dy/dx=v+x dv/dx
• Substitute → equation in v,x becomes separable
• Separate v and x integrate, replace v=y/x back
• Example (x²+y²)/xy homogeneous → (1+v²)/v
• Can also put x=v y if better
7. Linear Differential Equations - Integrating Factor IF
Linear first order DE form dy/dx + P(x) y = Q(x) where P,Q functions of x only. Method integrating factor IF = e^{∫P(x)dx}. Solution y·IF = ∫ Q·IF dx + C. Example dy/dx + 2y = e^x → P=2 IF=e^{∫2 dx}=e^{2x} y e^{2x}=∫ e^x e^{2x}dx=∫ e^{3x}dx=e^{3x}/3+C → y=e^x/3 + C e^{-2x}. Similarly dx/dy + P(y) x = Q(y) form linear in x, IF=e^{∫P(y)dy} solution x·IF=∫ Q·IF dy +C. This is most used for real life circuits, growth with limiting.
Linear first order: dy/dx + P(x) y = Q(x) IF=e^{∫P dx} Solution y·IF=∫ Q·IF dx +C. Example dy/dx+2y=e^x IF=e^{2x} y e^{2x}=∫ e^{3x}dx. Also dx/dy+P(y)x=Q(y) linear in x IF=e^{∫P(y)dy} x·IF=∫Q·IF dy+C.
• Form1: dy/dx + P(x) y = Q(x) P,Q functions of x
• IF = e^{∫P(x)dx}
• Solution: y·IF = ∫ Q·IF dx + C
• Example dy/dx+2y=e^x IF=e^{2x} → y e^{2x}=e^{3x}/3+C
• Form2: dx/dy + P(y) x = Q(y) linear in x
• IF = e^{∫P(y)dy} Solution x·IF=∫ Q·IF dy + C
• Used for circuits, growth with limiting, cooling
8. Summary - All Methods at a Glance
Summary of DE: Order highest derivative, Degree power of highest order when polynomial. General solution has constants = order, particular fixed. Formation eliminate constants differentiate n times. Variable separable dy/dx=g(x)h(y) → dy/h(y)=g(x)dx integrate. Homogeneous same degree put y=v x → v + x dv/dx separable. Linear dy/dx+P y=Q IF=e^{∫P dx} y·IF=∫ Q·IF dx+C. Other form dx/dy+P x=Q IF=e^{∫P dy}. Real life population growth radioactive decay circuits cooling. Choose method by seeing form.
Summary: Order highest derivative, Degree power. General constants=order, particular fixed. Formation eliminate constants n diff. Variable separable g(x)h(y) separate. Homogeneous same degree y=v x. Linear dy/dx+Py=Q IF=e^{∫P} y·IF=∫Q·IF. Real life growth decay circuit cooling.
• Order=highest derivative, Degree=power of highest order polynomial
• General constants=order, Particular fixed using y(x0)=y0
• Formation: n constants diff n times eliminate
• Variable Separable: dy/dx=g(x)h(y) → dy/h(y)=g(x)dx
• Homogeneous: same degree put y=v x → v+x dv/dx separable
• Linear: dy/dx+P(x)y=Q(x) IF=e^{∫P dx} y·IF=∫Q·IF dx+C
• Linear in x: dx/dy+P(y)x=Q(y) IF=e^{∫P dy} x·IF=∫Q·IF dy+C
• Real: Population growth, Decay, Cooling, Circuits, Medicine
• Real Life: Population dy/dt=k y → y=y0 e^{kt}, Decay dN/dt=-λ N, Cooling dT/dt=k(Ts-T), Circuit L di/dt+R i=V, Medicine concentration, Motion m d²x/dt²
• Order Degree: Order=highest derivative, Degree=power of highest order when polynomial, if derivative inside √/sin degree not defined
• General Particular: General constants=order family of curves, Particular fixed using y(x0)=y0 one curve
• Formation: n constants diff n times eliminate, constants=order, e.g. y=A sin x+B cos x → d²y/dx²+y=0
• Variable Separable: dy/dx=g(x)h(y) → dy/h(y)=g(x)dx → ∫dy/h(y)=∫g(x)dx, example x·y → y=C e^{x²/2}
• Homogeneous: Same degree f(x,y)/g(x,y) put y=v x → dy/dx=v+x dv/dx separable → replace v=y/x
• Linear: dy/dx+P(x)y=Q(x) IF=e^{∫P dx} y·IF=∫Q·IF dx+C, dx/dy+P(y)x=Q(y) IF=e^{∫P dy} x·IF=∫Q·IF dy+C
• Choose: Can separate?→Variable separable, Same degree?→Homogeneous y=vx, Linear in y or x?→IF method