Differential Equations — JEE Mains Mathematics

Formulas (2)

• Eliminating the constants of a family · Order = number of constants
  $$y=f(x;c_1,\dots,c_n)\ \Rightarrow\ F\big(x,y,y',\dots,y^{(n)}\big)=0$$
• Order, degree, and the equation a solution satisfies · Degree after clearing radicals
  $$y=x y'+\sqrt{1+y'^2}\ \Rightarrow\ (y-xy')^2=1+y'^2\ :\ \text{order }1,\ \text{degree }2$$

Watch out for (2)

• Use the conditions first
  → Eliminating the constants of a family
• Clear the radical before counting
  → Order, degree, and the equation a solution satisfies

Formulas (3)

• Separate, integrate, fix the constant · Separable form
  $$\frac{dy}{dx}=f(x)\,g(y)\ \Rightarrow\ \int\frac{dy}{g(y)}=\int f(x)\,dx+C$$
• When the slope depends on x alone · Direct integration
  $$\frac{dy}{dx}=f(x)\ \Rightarrow\ y=\int f(x)\,dx+C$$
• Substituting for x + y, and exact differentials · Substitution for a linear combination
  $$t=ax+by+c\ \Rightarrow\ \frac{dt}{dx}=a+b\,\frac{dy}{dx}$$

Watch out for (3)

• Constant before exponentiating
  → Separate, integrate, fix the constant
• Two conditions for a second-order equation
  → When the slope depends on x alone
• Differentiate the substitution fully
  → Substituting for x + y, and exact differentials

Formulas (2)

• Putting y = vx · The substitution
  $$y=vx:\quad v+x\frac{dv}{dx}=F(v)\ \Rightarrow\ \int\frac{dv}{F(v)-v}=\ln|x|+C$$
• When y = vx is not the first move · Shift of origin
  $$x=X+h,\ y=Y+k:\quad \frac{dY}{dX}=\frac{a_1X+b_1Y}{a_2X+b_2Y}$$

Watch out for (2)

• Return to y before using the point
  → Putting y = vx
• Parallel lines need a different substitution
  → When y = vx is not the first move

Formulas (2)

• Standard form and the integrating factor · Linear equation
  $$\frac{dy}{dx}+Py=Q\ \Rightarrow\ y\,e^{\int P\,dx}=\int Q\,e^{\int P\,dx}\,dx+C$$
• Integrating factors from trigonometric P · Common trigonometric factors
  $$P=k\tan x\ \Rightarrow\ \mu=\sec^kx,\qquad P=\frac{1}{1+x^2}\ \Rightarrow\ \mu=e^{\tan^{-1}x}$$

Watch out for (2)

• Standard form before the factor
  → Standard form and the integrating factor
• Sign of the tangent term
  → Integrating factors from trigonometric P

Formulas (2)

• The left side is already a derivative · Product rule in reverse
  $$u\,\frac{dy}{dx}+\frac{du}{dx}\,y=\frac{d}{dx}(uy)$$
• When P is the derivative of a logarithm · Log-derivative coefficient
  $$P=\frac{g'(x)}{g(x)}\ \Rightarrow\ e^{\int P\,dx}=g(x)$$

Watch out for (2)

• Divide only if it helps
  → The left side is already a derivative
• Check the derivative, do not integrate blindly
  → When P is the derivative of a logarithm

Formulas (3)

• Linear in x: dx/dy + P(y)x = Q(y) · Linear in x
  $$\frac{dx}{dy}+P(y)\,x=Q(y)\ \Rightarrow\ x\,e^{\int P\,dy}=\int Q\,e^{\int P\,dy}\,dy+C$$
• Bernoulli equations · Bernoulli substitution
  $$y'+Py=Qy^n,\ z=y^{1-n}\ \Rightarrow\ z'+(1-n)P\,z=(1-n)Q$$
• Substituting for a function of y · Substitution
  $$f'(y)\,\frac{dy}{dx}+P(x)\,f(y)=Q(x),\ u=f(y)\ \Rightarrow\ \frac{du}{dx}+Pu=Q$$

Watch out for (3)

• Integrate in y throughout
  → Linear in x: dx/dy + P(y)x = Q(y)
• The factor 1 - n
  → Bernoulli equations
• sin 2y becomes 2 tan y
  → Substituting for a function of y

Formulas (3)

• Limits that fix the constant · General solution with constant coefficient
  $$y'+ky=Q\ \Rightarrow\ y=y_p(x)+Ce^{-kx}$$
• Maxima, critical points and derivatives of the solution · Derivative from the equation
  $$\frac{dy}{dx}=Q(x)-P(x)\,y\ \ \text{(no need to differentiate the solution)}$$
• Integrating the solution: odd parts and areas · Symmetric interval
  $$\int_{-a}^{a}\big(g_{\text{odd}}+g_{\text{even}}\big)\,dx=2\int_0^{a}g_{\text{even}}\,dx$$

Watch out for (3)

• Which infinity
  → Limits that fix the constant
• Check the endpoint and the domain
  → Maxima, critical points and derivatives of the solution
• Split before integrating
  → Integrating the solution: odd parts and areas

Formulas (2)

• Differentiating an integral equation · Fundamental theorem
  $$f(x)=h(x)+\int_a^xg\big(t,f(t)\big)\,dt\ \Rightarrow\ f'(x)=h'(x)+g\big(x,f(x)\big),\ f(a)=h(a)$$
• Equations from limits and derivative rules · The limit as a derivative
  $$\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=2xf(x)-x^2f'(x)$$

Watch out for (2)

• The starting value is free
  → Differentiating an integral equation
• Watch the order in the denominator
  → Equations from limits and derivative rules

Formulas (2)

• Curves from tangent, normal and area conditions · Tangent at (x, y)
  $$Y-y=\frac{dy}{dx}(X-x):\quad X\text{-int}=x-\frac{y}{y'},\ \ Y\text{-int}=y-xy'$$
• Growth, decay and cooling · Newton's law of cooling
  $$\frac{dT}{dt}=-k(T-A)\ \Rightarrow\ T-A=(T_0-A)\,e^{-kt}$$

Watch out for (2)

• Midpoint and intercept are not the same
  → Curves from tangent, normal and area conditions
• The difference decays, not the temperature
  → Growth, decay and cooling