Picard’s Existence/Uniqueness Theorem; Lipschitz Condition

At a Glance

• Frequency: 1 sub-part across 1 of 13 years (2024)
• Priority tier: T4
• Marks (count): 15 (1)
• Average solve time: ~22 min
• Difficulty mix: medium 1
• Section: B | Dominant type: proof

Why This Chapter Matters

This atom appeared once (2024) in Section B for 15 marks. UPSC tests the theoretical side of ODEs: the precise statement of the Picard–Lindelöf theorem, verification of the Lipschitz condition for a given function, and the construction of Picard iterates. A question may ask for a full proof-sketch, or may ask you to apply the theorem to determine existence/uniqueness and compute the first few iterates for a specific IVP.

Minimum Theory

The initial-value problem

$y' = f(x,y), \quad y(x_0) = y_0$

Lipschitz condition

$f(x,y)$ satisfies a Lipschitz condition with respect to $y$ on a region $R$ if there exists a constant $K > 0$ (the Lipschitz constant) such that:

$|f(x,y_1) - f(x,y_2)| \leq K|y_1 - y_2| \quad \text{for all }(x,y_1),(x,y_2)\in R$

Sufficient condition: If $\partial f/\partial y$ exists and $\left|\dfrac{\partial f}{\partial y}\right| \leq K$ on $R$, then $f$ satisfies the Lipschitz condition with constant $K$ (by the Mean Value Theorem).

Picard–Lindelöf (Picard’s Existence and Uniqueness) Theorem

Hypotheses. Let $R = \{(x,y): |x-x_0|\leq a,\; |y-y_0|\leq b\}$. Suppose:

1. $f(x,y)$ is continuous on $R$, so $|f(x,y)| \leq M$ for some $M > 0$ on $R$.
2. $f$ satisfies the Lipschitz condition in $y$ on $R$ with constant $K$.

Conclusion. The IVP $y' = f(x,y)$, $y(x_0) = y_0$ has a unique solution on the interval $|x - x_0| \leq h$, where

$h = \min\!\left(a,\, \frac{b}{M}\right)$

Picard iteration (successive approximations)

Define the sequence:

$y_0(x) = y_0$

$y_{n+1}(x) = y_0 + \int_{x_0}^{x} f\!\left(t,\, y_n(t)\right) dt, \quad n = 0, 1, 2, \ldots$

Under the hypotheses, $\{y_n(x)\}$ converges uniformly to the unique solution on $|x-x_0|\leq h$.

Error bound

$|y(x) - y_n(x)| \leq \frac{MK^n h^{n+1}}{(n+1)!} \cdot \frac{1}{1 - ?}$

A useful simpler bound: the iterates converge because the error is of order $(Kh)^n/(n!)$ which tends to zero.

Proof sketch (for 15-mark questions)

1. Equivalence: The IVP $y' = f, y(x_0)=y_0$ is equivalent to the integral equation $y = y_0 + \int_{x_0}^x f(t,y(t))\,dt$.
2. Picard operator: Define $T[y](x) = y_0 + \int_{x_0}^x f(t,y(t))\,dt$; a solution is a fixed point of $T$.
3. Successive approximations stay in $R$: Show $|y_1(x)-y_0|\leq M|x-x_0|\leq Mh\leq b$, so $y_1$ stays in $R$; by induction all $y_n$ stay in $R$.
4. Uniform convergence: Show $|y_{n+1}-y_n| \leq \dfrac{MK^n h^{n+1}}{(n+1)!}$ by induction using the Lipschitz condition; this is the general term of the exponential series, which converges.
5. Limit is a solution: Passing the limit inside the integral (justified by uniform convergence) shows the limit $y = \lim y_n$ satisfies the integral equation.
6. Uniqueness: If $y$ and $z$ are two solutions, then $|y-z| \leq K\int_{x_0}^x|y-z|\,dt$; by Grönwall’s inequality (or repeated application), $|y-z| = 0$.

Question Archetypes

picard-iterate-and-state (1 question; 2024)

Recognition Cues

• The question uses the phrase “Picard’s theorem”, “existence and uniqueness”, or “Lipschitz condition”.
• An explicit IVP is given for which iterates are to be computed.
• You may be asked to verify the hypothesis, state the conclusion (interval of existence), and compute $y_1$, $y_2$, (sometimes $y_3$).
• The function $f$ is typically simple (e.g., $f = xy$, $f = x+y$, $f = y$).

Solution Template

1. State the Picard–Lindelöf theorem precisely (hypotheses and conclusion).
2. Verify continuity of $f$ on $R$ and compute $M = \max_R|f|$.
3. Verify the Lipschitz condition (compute $\partial f/\partial y$; bound it by $K$).
4. State the interval of existence $|x-x_0|\leq h = \min(a, b/M)$.
5. Compute $y_0$, $y_1$, $y_2$ (and $y_3$ if marks warrant) by the Picard iteration formula.
6. Identify the pattern if the iterates form a recognisable series (e.g., partial sums of $e^x$).

Worked Example

2024 Paper 1, 2024-P1-Q7b (15 marks)

State Picard’s existence and uniqueness theorem. Verify its hypotheses for the IVP $y' = x + y, \quad y(0) = 0,$ and compute the first three Picard iterates $y_1, y_2, y_3$.

Step 1. Statement of the theorem.

Let $R: |x|\leq a$, $|y|\leq b$. If $f(x,y)$ is continuous on $R$ with $|f|\leq M$ and satisfies $|f(x,y_1)-f(x,y_2)|\leq K|y_1-y_2|$ on $R$, then $y'=f(x,y)$, $y(0)=0$ has a unique solution on $|x|\leq h = \min(a, b/M)$.

Step 2. Verify hypotheses for $f(x,y) = x+y$.

Take $R: |x|\leq a$, $|y|\leq b$ (e.g., $a=b=1$).

$f = x+y$ is a polynomial, hence continuous on $R$. On $R$: $|f| = |x+y| \leq a + b = M$.

Lipschitz condition: $|f(x,y_1)-f(x,y_2)| = |y_1-y_2|$, so $K = 1$.

Interval of existence: $h = \min\!\left(1,\, \dfrac{1}{1+1}\right) = \dfrac{1}{2}$.

Step 3. Picard iterates. Set $x_0 = 0$, $y_0 = 0$.

$y_0(x) = 0$

$y_1(x) = 0 + \int_0^x (t + y_0(t))\,dt = \int_0^x t\,dt = \frac{x^2}{2}$

$y_2(x) = 0 + \int_0^x \left(t + y_1(t)\right)dt = \int_0^x \left(t + \frac{t^2}{2}\right)dt = \frac{x^2}{2} + \frac{x^3}{6}$

$y_3(x) = 0 + \int_0^x \left(t + y_2(t)\right)dt = \int_0^x \left(t + \frac{t^2}{2} + \frac{t^3}{6}\right)dt = \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24}$

Step 4. Pattern and limiting solution.

The iterates are partial sums of the series for $e^x - x - 1$:

$y_n(x) \to y(x) = e^x - x - 1$

which can be verified: $(e^x - x - 1)' = e^x - 1 = (e^x - x - 1) + x$, and $y(0) = 0$. ✓

$\boxed{y_1 = \frac{x^2}{2},\quad y_2 = \frac{x^2}{2}+\frac{x^3}{6},\quad y_3 = \frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24};\quad y = e^x - x - 1}$

Common Traps

• Stating the Lipschitz condition as $|f(x_1,y)-f(x_2,y)|\leq K|x_1-x_2|$ (wrong variable — it must be with respect to $y$).
• Computing $y_1(x) = \int_0^x f(t, y_0)\,dt$ but substituting $x$ instead of $t$ inside the integrand.
• Forgetting the additive $y_0$ in the iteration formula, writing $y_{n+1} = \int f$ instead of $y_{n+1} = y_0 + \int f$.
• Omitting the verification that iterates remain in $R$ (examiners expect at least a brief acknowledgement).

Marks-Aware Writing

This is a 15-mark Section B proof+application question. Examiners expect:

1. Precise theorem statement — hypotheses ($R$, continuity, $M$, Lipschitz, $K$) and conclusion ($h$, unique solution) (4 marks).
2. Hypothesis verification — continuity confirmed, $M$ computed, Lipschitz checked with explicit $K$ (3 marks).
3. Interval of existence computed (1 mark).
4. Three Picard iterates computed step by step, with integral shown for each (6 marks, 2 per iterate).
5. Pattern / exact solution identified and verified (1 mark).

Do not skip showing the integral at each step — substituting $y_n$ into the integrand must be written out.

Practice Set

Only one historical question on this atom (shown above).