Differential Equations — MHT-CET Maths

Formulas (9)

• Differential Equation Terminology · The master link
  $$\text{order of the ODE} \;=\; \text{number of independent arbitrary constants in its general solution}$$
• Order = Order of the Highest Derivative Present · Order
  $$\text{order} = \text{the order of the highest derivative appearing in the equation}$$
• Degree = Power of the Highest Derivative After Clearing Radicals · Degree
  $$\text{degree} = \text{power of the highest-order derivative, once the equation is polynomial in its derivatives}$$
• When Degree Is Undefined (Derivative Inside a Transcendental) · Degree-undefined criterion
  $$\text{degree undefined} \iff \text{a derivative sits inside a transcendental (}\log,\ \sin,\ \cos,\ e^{(\cdot)}\text{)}$$
• Collapse Redundant Arbitrary Constants Before Counting Order · Constant-absorption identity
  $$C_3\,e^{\,x + C_4} = \big(C_3 e^{C_4}\big)e^{x} = B\,e^{x}$$
• Formation: n Independent Constants ⇒ Order-n Differential Equation · Formation order
  $$n \text{ independent arbitrary constants} \;\Longrightarrow\; \text{differential equation of order } n$$
• Forming the Differential Equation of a Curve Family · Elimination recipe
  $$\text{differentiate } n \text{ times} \;\to\; \text{solve for the constants} \;\to\; \text{substitute back to eliminate them}$$
• Forming the Differential Equation of Circles and Parabolas · Two workhorses
  $$x^2 + y^2 = 2ax \;(\text{circle}) \qquad x^2 = 4ay \;(\text{parabola, axis } +Y)$$
• Verifying a Solution and Identifying Its Family · Parametric derivative
  $$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} \qquad \dfrac{d^2y}{dx^2} = \dfrac{1}{dx/dt}\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)$$

Watch out for (17)

• Order and degree are separate labels
  → Differential Equation Terminology
• "Number of constants" means INDEPENDENT constants
  → Differential Equation Terminology
• A power on the top derivative is DEGREE, never order
  → Order = Order of the Highest Derivative Present
• Clear radicals BEFORE you read the degree
  → Degree = Power of the Highest Derivative After Clearing Radicals
• Raise to the LCM of the fractional exponents
  → Degree = Power of the Highest Derivative After Clearing Radicals
• Seeing a first power does NOT mean degree 1
  → When Degree Is Undefined (Derivative Inside a Transcendental)
• Order survives; only degree dies
  → When Degree Is Undefined (Derivative Inside a Transcendental)
• $e^{x + C}$ hides a constant, it does not add one
  → Collapse Redundant Arbitrary Constants Before Counting Order
• Only INDEPENDENT constants count
  → Collapse Redundant Arbitrary Constants Before Counting Order
• Collapse constants BEFORE fixing the order
  → Formation: n Independent Constants ⇒ Order-n Differential Equation
• A fixed point removes a constant
  → Formation: n Independent Constants ⇒ Order-n Differential Equation
• Eliminate the CONSTANT, not the known function
  → Forming the Differential Equation of a Curve Family
• Differentiate ONCE per constant — no more, no less
  → Forming the Differential Equation of a Curve Family
• Translate the geometry into the RIGHT free constants
  → Forming the Differential Equation of Circles and Parabolas
• Mind the sign when substituting the eliminated constant
  → Forming the Differential Equation of Circles and Parabolas
• Convert parametric derivatives correctly
  → Verifying a Solution and Identifying Its Family
• Identify the conic from the SIMPLIFIED solution
  → Verifying a Solution and Identifying Its Family

Formulas (7)

• The Separate-Then-Integrate Idea · Separable form and its solution
  $$\dfrac{dy}{dx} = f(x)\,g(y) \;\Longrightarrow\; \int \dfrac{dy}{g(y)} = \int f(x)\,dx + c$$
• Basic Separation and Integrating Both Sides · Standard integrals used after separating
  $$\int \dfrac{dy}{y} = \log y + c,\qquad \int \dfrac{dx}{x^2} = -\dfrac{1}{x} + c$$
• Applying an Initial Condition (Particular Solutions) · General → particular via the condition
  $$y = \Phi(x, c),\qquad y(x_0) = y_0 \;\Rightarrow\; c = c_0 \;\Rightarrow\; y = \Phi(x, c_0)$$
• Separables in Disguise — Logs and Exponential Right Sides · Exponentiate to separate; the eˣ(f + f′) trick
  $$\log\!\Big(\tfrac{dy}{dx}\Big) = ax+by \;\Rightarrow\; e^{-by}\,dy = e^{ax}\,dx,\qquad \int e^x\big(f(x)+f'(x)\big)\,dx = e^x f(x) + c$$
• Trigonometric-Product Separables · The log-integrals you reach for
  $$\int \dfrac{\cos x}{1+\sin x}\,dx = \log(1+\sin x) + c,\qquad \int \tan y\,dy = -\log\cos y + c = \log\sec y + c$$
• Rational Separables — arctan, arcsin, and Families of Circles · arctan and the circle-producing integral
  $$\int \dfrac{dy}{1+y^2} = \tan^{-1}y + c,\qquad \int \dfrac{y\,dy}{\sqrt{k^2-y^2}} = -\sqrt{k^2-y^2} + c$$
• Direct Integration — dy/dx = f(x) and Slope-of-Curve Problems · Pure x-side integration
  $$\dfrac{dy}{dx} = f(x) \;\Longrightarrow\; y = \int f(x)\,dx + c$$

Watch out for (14)

• One arbitrary constant, and add it at the integration step
  → The Separate-Then-Integrate Idea
• You cannot divide by a factor that might be zero
  → The Separate-Then-Integrate Idea
• Absorb the constant as $\log c$, not $+c$, when both sides are logs
  → Basic Separation and Integrating Both Sides
• $x\dfrac{dy}{dx} = 2y$ is a parabola family, not a linear one
  → Basic Separation and Integrating Both Sides
• Don't forget the $+c$ BEFORE applying the initial condition
  → Applying an Initial Condition (Particular Solutions)
• Watch the $\log$ → product conversion
  → Applying an Initial Condition (Particular Solutions)
• Take logs / exponentials to unlock separation
  → Separables in Disguise — Logs and Exponential Right Sides
• Spot the $\int e^x(f+f')\,dx = e^x f$ pattern
  → Separables in Disguise — Logs and Exponential Right Sides
• Apply product-to-sum BEFORE trying to separate
  → Trigonometric-Product Separables
• Signs of the trig log-integrals
  → Trigonometric-Product Separables
• Write the arctan constant as $\tan^{-1}c$, then use the subtraction formula
  → Rational Separables — arctan, arcsin, and Families of Circles
• Identify the circle's centre-axis and radius carefully
  → Rational Separables — arctan, arcsin, and Families of Circles
• Simplify the RHS before integrating
  → Direct Integration — dy/dx = f(x) and Slope-of-Curve Problems
• Divide the polynomial before integrating a rational $f(x)$
  → Direct Integration — dy/dx = f(x) and Slope-of-Curve Problems

Formulas (6)

• Recognizing a Homogeneous Differential Equation · Homogeneity test
  $$f(\lambda x, \lambda y) = \lambda^n f(x,y) \quad\Longleftrightarrow\quad \dfrac{dy}{dx} = g\!\left(\dfrac{y}{x}\right)$$
• The y = vx Substitution · Homogeneous substitution
  $$y = vx \;\Rightarrow\; \dfrac{dy}{dx} = v + x\dfrac{dv}{dx} \;\Rightarrow\; \dfrac{dv}{g(v) - v} = \dfrac{dx}{x}$$
• Worked Homogeneous Equations and Initial Conditions · Particular solution from an IC
  $$\text{integrate} \;\to\; \text{back-substitute } v=\tfrac{y}{x} \;\to\; \text{plug the point to find } c$$
• Homogeneous Curves Through a Point (Trig Ratio Slopes) · Trig-ratio homogeneous slopes
  $$\dfrac{dy}{dx} = \dfrac{y}{x} + h\!\left(\dfrac{y}{x}\right) \;\xrightarrow{\,y=vx\,}\; \dfrac{dv}{h(v)} = \dfrac{dx}{x}$$
• Log-Form Homogeneous Equations (v = y/x) · The log-form integral
  $$\int\dfrac{dv}{v\,\log v} = \log(\log v) \;\Rightarrow\; \log\dfrac{y}{x} = c\,x$$
• Reducible to Separable via v = x + y (or v = ax + by) · Linear-argument substitution
  $$v = x + y \;\Rightarrow\; \dfrac{dv}{dx} = 1 + \dfrac{dy}{dx} \;\Rightarrow\; \dfrac{dv}{1 + f(v)} = dx$$

Watch out for (13)

• A stray constant breaks homogeneity
  → Recognizing a Homogeneous Differential Equation
• Same degree top and bottom is the fast check
  → Recognizing a Homogeneous Differential Equation
• dy/dx is v + x·dv/dx, not just dv/dx
  → The y = vx Substitution
• Substitute v = y/x back at the very end
  → The y = vx Substitution
• Use the initial condition only after back-substituting
  → Worked Homogeneous Equations and Initial Conditions
• Track the sign of g(v) − v
  → Worked Homogeneous Equations and Initial Conditions
• The bare y/x cancels — don't integrate it
  → Homogeneous Curves Through a Point (Trig Ratio Slopes)
• Feed the initial point to find c — always
  → Homogeneous Curves Through a Point (Trig Ratio Slopes)
• The integral is log(log v), not log v
  → Log-Form Homogeneous Equations (v = y/x)
• It's cx, not cy — check which variable the constant multiplies
  → Log-Form Homogeneous Equations (v = y/x)
• v = x + y gives dv/dx = 1 + dy/dx — keep the +1
  → Reducible to Separable via v = x + y (or v = ax + by)
• For v = ax + by, the coefficient rides through
  → Reducible to Separable via v = x + y (or v = ax + by)
• Substitute v = x + y back at the end
  → Reducible to Separable via v = x + y (or v = ax + by)

Formulas (8)

• Recognizing the Standard Linear Form · Standard linear form
  $$\dfrac{dy}{dx} + P(x)\,y = Q(x)$$
• The Integrating Factor and the Solution Formula · Integrating factor and general solution
  $$\text{IF} = e^{\int P(x)\,dx}, \qquad y\cdot\text{IF} = \int Q(x)\cdot\text{IF}\,dx + c$$
• Simple Integrating Factors · Common integrating factors
  $$P=\dfrac{n}{x}\Rightarrow\text{IF}=x^{n}, \quad P=k\Rightarrow\text{IF}=e^{kx}, \quad P=-\tan x\Rightarrow\text{IF}=\cos x$$
• Tricky Integrating Factors · A tricky IF built by partial fractions
  $$P = -\dfrac{2}{x} + \dfrac{1}{x-1} \;\Rightarrow\; \int P\,dx = \log\dfrac{x-1}{x^2} \;\Rightarrow\; \text{IF} = \dfrac{x-1}{x^2}$$
• Linear in x — Swap the Roles of x and y · Linear in x (reciprocal form)
  $$\dfrac{dx}{dy} + P(y)\,x = Q(y), \qquad \text{IF} = e^{\int P(y)\,dy}$$
• Bernoulli Equations — Substitute to Linearize · Bernoulli substitution
  $$\dfrac{dy}{dx} + Py = Qy^{n} \;\xrightarrow{\;v = y^{1-n}\;}\; \dfrac{dv}{dx} + (1-n)Pv = (1-n)Q$$
• Exact Equations by d(·)-Grouping · Exact differentials to spot
  $$x\,dy + y\,dx = d(xy), \qquad \dfrac{x\,dy - y\,dx}{y^2} = d\!\left(\dfrac{x}{y}\right)$$
• Direct Integration and Reduction of Order · Reduction of order (integrate twice)
  $$\dfrac{d^2y}{dx^2} = g(x) \;\Rightarrow\; \dfrac{dy}{dx} = \int g(x)\,dx + c_1 \;\Rightarrow\; y = \int\!\left(\int g\,dx\right)dx + c_1 x + c_2$$

Watch out for (16)

• Read $P$ only AFTER making the $\dfrac{dy}{dx}$ coefficient $1$
  → Recognizing the Standard Linear Form
• A $y^2$, $\sqrt{y}$, or $1/y$ means it is NOT linear (yet)
  → Recognizing the Standard Linear Form
• The left side is $\dfrac{d}{dx}(y\cdot\text{IF})$ — do not re-differentiate the product
  → The Integrating Factor and the Solution Formula
• One arbitrary constant only, added at the integration step
  → The Integrating Factor and the Solution Formula
• $e^{\log f(x)} = f(x)$ — simplify the exponential of a log
  → Simple Integrating Factors
• Watch the sign of $P$ in the exponential
  → Simple Integrating Factors
• Split $P$ before integrating a rational coefficient
  → Tricky Integrating Factors
• Do not stop at $\int P\,dx$ — exponentiate it
  → Tricky Integrating Factors
• If $y$ is tangled, check whether $x$ is linear before giving up
  → Linear in x — Swap the Roles of x and y
• After flipping, integrate with respect to $y$, not $x$
  → Linear in x — Swap the Roles of x and y
• Divide by $y^{n}$ BEFORE substituting
  → Bernoulli Equations — Substitute to Linearize
• Spot the lone $y^{n}$ — it is not a linear ODE
  → Bernoulli Equations — Substitute to Linearize
• Mind the sign and denominator of the quotient differentials
  → Exact Equations by d(·)-Grouping
• Try grouping before reaching for an integrating factor
  → Exact Equations by d(·)-Grouping
• Apply the slope condition after the FIRST integration
  → Direct Integration and Reduction of Order
• Divide out the leading factor before integrating
  → Direct Integration and Reduction of Order

Formulas (7)

• The Modelling Step — Rate Proportional to Quantity · Rate proportional to quantity
  $$\dfrac{dP}{dt} = kP \quad\Longrightarrow\quad \log P = kt + c$$
• The Exponential Solution P = P0 e^{kt} and Finding k · Exponential growth/decay solution
  $$P = P_0\,e^{kt}, \qquad k = \dfrac{1}{t_1}\log\!\dfrac{P_1}{P_0}$$
• Population and Bacteria — Doubling Time and Percentage Growth · Doubling growth
  $$e^{kT} = 2 \quad\Longrightarrow\quad P = P_0\,2^{\,t/T}$$
• Radioactive Decay and Half-Life · Half-life rate constant
  $$k = \dfrac{\log 2}{h}, \qquad m = m_0\left(\tfrac12\right)^{t/h}$$
• Continuous Compounding of Money · Continuous compounding
  $$A = P\,e^{rt}, \qquad e^{rT} = 2 \text{ if the principal doubles in } T$$
• Moisture Loss and General First-Order Rate Models · Fraction-lost time (pure decay)
  $$\log\dfrac{P}{N} = -kt, \qquad k = \log 2 \text{ if half is lost in unit time}$$
• Special-Rate Models — Square-Root and Surface-Area Decay · Square-root and surface-area models
  $$\dfrac{dx}{dt} = -k\sqrt{x} \Rightarrow 2\sqrt{x} = -kt + c; \qquad \dfrac{dV}{dt} = -kS \Rightarrow \dfrac{dr}{dt} = -k$$

Watch out for (14)

• Decay carries a negative sign
  → The Modelling Step — Rate Proportional to Quantity
• 'Proportional to' is not 'equal to'
  → The Modelling Step — Rate Proportional to Quantity
• Cancel $P_0$ by dividing — don't solve for k first
  → The Exponential Solution P = P0 e^{kt} and Finding k
• The extra time is measured from the start
  → The Exponential Solution P = P0 e^{kt} and Finding k
• 'Doubles' means the ratio is 2, not '+2'
  → Population and Bacteria — Doubling Time and Percentage Growth
• Turn a percentage into a factor before touching k
  → Population and Bacteria — Doubling Time and Percentage Growth
• The initial decay rate is negative
  → Radioactive Decay and Half-Life
• Count half-lives only when time is a whole multiple
  → Radioactive Decay and Half-Life
• Convert the % rate to a decimal
  → Continuous Compounding of Money
• Continuous compounding uses $e^{rt}$, not $(1+r)^t$
  → Continuous Compounding of Money
• '99% lost' means the fraction LEFT is 0.01
  → Moisture Loss and General First-Order Rate Models
• The constant term needs factoring before you separate
  → Moisture Loss and General First-Order Rate Models
• $\int \dfrac{dx}{\sqrt x} = 2\sqrt x$, not $\log\sqrt x$
  → Special-Rate Models — Square-Root and Surface-Area Decay
• Surface-area evaporation makes the RADIUS linear
  → Special-Rate Models — Square-Root and Surface-Area Decay

Formulas (4)

• The Cooling Model — Rate Proportional to Temperature Excess · Newton's law of cooling
  $$\dfrac{d\theta}{dt} = -k(\theta - \theta_s), \qquad k > 0$$
• Solving the Cooling Equation — Log Form and Exponential Form · Log form and its exponential solution
  $$\log(\theta - \theta_s) = -kt + c \qquad\Longleftrightarrow\qquad \theta - \theta_s = (\theta_0 - \theta_s)\,e^{-kt}$$
• Two-Stage Cooling — Fix the Rate, Then Predict · Ratio of excesses over two intervals
  $$\dfrac{\theta_2 - \theta_s}{\theta_1 - \theta_s} = e^{-k(t_2 - t_1)}$$
• The (Ratio)ⁿ Shortcut for Equal Time-Steps · Geometric decay of the excess over n equal steps
  $$\theta_n - \theta_s = (\theta_0 - \theta_s)\,r^{\,n}, \qquad r = e^{-k\Delta t}$$

Watch out for (8)

• Always work with the excess $\theta - \theta_s$, not $\theta$
  → The Cooling Model — Rate Proportional to Temperature Excess
• The minus sign and $k>0$ together mean cooling
  → The Cooling Model — Rate Proportional to Temperature Excess
• Take the log of the EXCESS, not the temperature
  → Solving the Cooling Equation — Log Form and Exponential Form
• Here $\log$ means natural log
  → Solving the Cooling Equation — Log Form and Exponential Form
• Subtract the surrounding temperature BEFORE forming the ratio
  → Two-Stage Cooling — Fix the Rate, Then Predict
• Match the exponent to the number of equal intervals
  → Two-Stage Cooling — Fix the Rate, Then Predict
• Equal steps ⇒ geometric ratio of the EXCESSES
  → The (Ratio)ⁿ Shortcut for Equal Time-Steps
• Count n as total time ÷ interval, then raise the ratio to that power
  → The (Ratio)ⁿ Shortcut for Equal Time-Steps