NUM8ERS Tutoring Center Dubai logo – you can count on us
Book Now
Sign In

Unit 7.2 – Verifying Solutions for Differential Equations

AP® Calculus AB & BC | Checking if Functions Satisfy Differential Equations

Why This Matters: Before solving differential equations, you need to understand what it means for a function to BE a solution! Verification is the process of checking whether a given function satisfies a differential equation. This is a crucial skill—it appears on virtually every AP® exam and helps you understand the relationship between functions and their derivatives. Master verification and you'll gain deep insight into how differential equations work!

🎯 What Does It Mean to Verify?

VERIFICATION DEFINITION

To verify that a function \(y = f(x)\) is a solution to a differential equation means to show that when you substitute the function and its derivatives into the equation, both sides are equal.

The Process:
  1. Find the necessary derivative(s) of \(y\)
  2. Substitute \(y\) and its derivative(s) into the differential equation
  3. Simplify both sides
  4. Check if both sides are identical

📝 Key Insight: Verification is essentially checking that the equation is TRUE when you plug in the proposed solution!

📋 The Verification Process

Step-by-Step Procedure

Standard Verification Steps:

  1. Identify what's given:
    • The differential equation
    • The proposed solution \(y = f(x)\)
  2. Find required derivatives:
    • If DE has \(\frac{dy}{dx}\), find \(y'\)
    • If DE has \(\frac{d^2y}{dx^2}\), find \(y''\) also
  3. Substitute into DE:
    • Replace \(y\) with the given function
    • Replace derivatives with calculated values
  4. Simplify both sides:
    • Combine like terms
    • Factor if helpful
  5. Compare and conclude:
    • If LHS = RHS, it IS a solution ✓
    • If LHS ≠ RHS, it is NOT a solution ✗

📖 Comprehensive Worked Examples

Example 1: Basic Verification

Problem: Verify that \(y = e^{3x}\) is a solution to \(\frac{dy}{dx} = 3y\)

Solution:

Step 1: Identify given information

  • Differential equation: \(\frac{dy}{dx} = 3y\)
  • Proposed solution: \(y = e^{3x}\)

Step 2: Find derivative

\[ \frac{dy}{dx} = \frac{d}{dx}[e^{3x}] = 3e^{3x} \]

Step 3: Substitute into DE

Left side: \(\frac{dy}{dx} = 3e^{3x}\)

Right side: \(3y = 3(e^{3x}) = 3e^{3x}\)

Step 4: Compare

\[ 3e^{3x} = 3e^{3x} \quad ✓ \]

✓ VERIFIED: \(y = e^{3x}\) IS a solution!

Example 2: Polynomial Solution

Problem: Verify that \(y = x^2 + 2x - 1\) is a solution to \(\frac{dy}{dx} = 2x + 2\)

Solution:

Step 1: Given

  • DE: \(\frac{dy}{dx} = 2x + 2\)
  • Proposed: \(y = x^2 + 2x - 1\)

Step 2: Find derivative

\[ \frac{dy}{dx} = 2x + 2 \]

Step 3-4: Substitute and compare

Left side: \(\frac{dy}{dx} = 2x + 2\)

Right side: \(2x + 2\)

\[ 2x + 2 = 2x + 2 \quad ✓ \]

✓ VERIFIED: It IS a solution!

Example 3: More Complex - Product

Problem: Verify that \(y = x^2 e^x\) is a solution to \(\frac{dy}{dx} = y + 2xe^x\)

Solution:

Step 1: Given

  • DE: \(\frac{dy}{dx} = y + 2xe^x\)
  • Proposed: \(y = x^2 e^x\)

Step 2: Find derivative (use product rule)

\[ \frac{dy}{dx} = \frac{d}{dx}[x^2 e^x] = 2xe^x + x^2e^x = e^x(2x + x^2) \]

Step 3: Substitute into DE

Left side: \(\frac{dy}{dx} = e^x(2x + x^2) = x^2e^x + 2xe^x\)

Right side: \(y + 2xe^x = x^2e^x + 2xe^x\)

Step 4: Compare

\[ x^2e^x + 2xe^x = x^2e^x + 2xe^x \quad ✓ \]

✓ VERIFIED: \(y = x^2e^x\) IS a solution!

Example 4: Second-Order DE

Problem: Verify that \(y = \sin(2x)\) is a solution to \(\frac{d^2y}{dx^2} + 4y = 0\)

Solution:

Step 1: Given

  • DE: \(\frac{d^2y}{dx^2} + 4y = 0\)
  • Proposed: \(y = \sin(2x)\)

Step 2: Find first derivative

\[ \frac{dy}{dx} = 2\cos(2x) \]

Find second derivative

\[ \frac{d^2y}{dx^2} = -4\sin(2x) \]

Step 3: Substitute into DE

\[ \frac{d^2y}{dx^2} + 4y = -4\sin(2x) + 4\sin(2x) = 0 \]

Step 4: Compare

\[ 0 = 0 \quad ✓ \]

✓ VERIFIED: \(y = \sin(2x)\) IS a solution!

Example 5: NOT a Solution

Problem: Check if \(y = x^2\) is a solution to \(\frac{dy}{dx} = 3x\)

Solution:

Step 1-2: Find derivative

Given: \(y = x^2\)

\[ \frac{dy}{dx} = 2x \]

Step 3-4: Substitute and compare

Left side: \(\frac{dy}{dx} = 2x\)

Right side: \(3x\)

\[ 2x \neq 3x \quad ✗ \]

✗ NOT A SOLUTION: \(y = x^2\) does NOT satisfy the equation!

🎲 Verifying with Initial Conditions

Two-Part Verification for IVPs:

For Initial Value Problems, you must verify TWO things:

  1. The function satisfies the differential equation
  2. The function satisfies the initial condition

Example 6: IVP Verification

Problem: Verify that \(y = 2e^{3x}\) is a solution to the IVP: \(\frac{dy}{dx} = 3y\), \(y(0) = 2\)

Solution:

Part 1: Verify DE

Given: \(y = 2e^{3x}\)

Derivative: \(\frac{dy}{dx} = 6e^{3x}\)

Check: \(3y = 3(2e^{3x}) = 6e^{3x}\)

\[ \frac{dy}{dx} = 3y \quad ✓ \]

Part 2: Verify initial condition

Check \(y(0) = 2\):

\[ y(0) = 2e^{3(0)} = 2e^0 = 2(1) = 2 \quad ✓ \]

✓ FULLY VERIFIED: Satisfies both DE and initial condition!

🔍 Common Types of Verifications

Verification Scenarios

What You Might Need to Verify
Type What to Check Example
General Solution Function with constant \(C\) \(y = Ce^{kx}\)
Particular Solution Specific function (no \(C\)) \(y = 3e^{2x}\)
Initial Value Problem DE + initial condition \(y' = 2y, y(0) = 5\)
Multiple Functions Test each one Is \(y = x\) or \(y = x^2\) a solution?

💡 Essential Tips & Strategies

✅ Verification Success Tips:

  • Write everything out: Don't skip steps—show your work clearly
  • Use proper notation: Keep \(\frac{dy}{dx}\) and \(y\) distinct
  • Check your derivative: Most errors happen here!
  • Substitute completely: Replace ALL instances of \(y\) and derivatives
  • Simplify fully: Both sides should be in simplest form before comparing
  • For IVPs: Always verify both the DE and initial condition
  • Work neatly: Clear work prevents errors

🔥 Common Calculus Rules to Remember:

  • Product Rule: \((uv)' = u'v + uv'\)
  • Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
  • Exponential: \(\frac{d}{dx}[e^{kx}] = ke^{kx}\)
  • Trig: \(\frac{d}{dx}[\sin(kx)] = k\cos(kx)\)
  • Trig: \(\frac{d}{dx}[\cos(kx)] = -k\sin(kx)\)

❌ Common Mistakes to Avoid

  • Mistake 1: Forgetting the chain rule when differentiating
  • Mistake 2: Not substituting the derivative correctly
  • Mistake 3: Comparing before simplifying fully
  • Mistake 4: Sign errors (especially with trig derivatives)
  • Mistake 5: Forgetting to verify initial condition for IVPs
  • Mistake 6: Confusing \(y\) with \(\frac{dy}{dx}\)
  • Mistake 7: Arithmetic errors in simplification
  • Mistake 8: Not showing enough work (on exams!)
  • Mistake 9: Assuming it's a solution without checking
  • Mistake 10: For general solutions, forgetting the constant \(C\) matters

📝 Practice Problems

Verify the following:

  1. Is \(y = 4e^{-2x}\) a solution to \(\frac{dy}{dx} = -2y\)?
  2. Is \(y = x^3 - 3x\) a solution to \(\frac{dy}{dx} = 3x^2 - 3\)?
  3. Is \(y = \cos(3x)\) a solution to \(\frac{d^2y}{dx^2} + 9y = 0\)?
  4. Does \(y = e^x + 1\) satisfy \(\frac{dy}{dx} = y\) with \(y(0) = 2\)?

Answers:

  1. YES: \(y' = -8e^{-2x}\) and \(-2y = -8e^{-2x}\) ✓
  2. YES: \(y' = 3x^2 - 3\) ✓
  3. YES: \(y'' = -9\cos(3x)\), so \(y'' + 9y = 0\) ✓
  4. YES for DE, YES for IC: \(y' = e^x = y\) ✓, and \(y(0) = e^0 + 1 = 2\) ✓

✏️ AP® Exam Success Tips

What AP® Graders Look For:

  • Show the derivative: Write it out explicitly
  • Substitute clearly: Show what you're plugging in
  • Show simplification: Don't jump to conclusion
  • State conclusion: "Therefore, \(y\) is a solution" or "is not a solution"
  • For IVPs: Verify BOTH parts (DE and IC)
  • Use proper notation: \(\frac{dy}{dx}\) not \(dy/dx\) or \(y'\)
  • Be organized: Clear, sequential work earns points

💯 Exam Strategy:

  1. Read carefully—what are you verifying?
  2. Find derivatives FIRST (before substituting)
  3. Write out the substitution step by step
  4. Simplify until both sides match (or don't)
  5. State your conclusion explicitly
  6. If time permits, double-check your derivative

⚡ Quick Reference Guide

VERIFICATION CHECKLIST

The 5-Step Process:

  1. ✓ Identify: What's the DE? What's the proposed solution?
  2. ✓ Differentiate: Find \(y'\) (and \(y''\) if needed)
  3. ✓ Substitute: Replace \(y\) and derivatives in DE
  4. ✓ Simplify: Work both sides to simplest form
  5. ✓ Compare: LHS = RHS? State conclusion!

For Initial Value Problems:

  • Step 6: Verify initial condition \(y(x_0) = y_0\)
  • Both must be true for complete verification!

Master Verification! Verifying solutions means checking that a function satisfies a differential equation. The process: (1) Find all necessary derivatives, (2) Substitute the function and its derivatives into the DE, (3) Simplify both sides completely, (4) Compare—if both sides are equal, it's a solution; if not, it isn't. For Initial Value Problems, you must verify TWO things: the function satisfies the differential equation AND it satisfies the initial condition \(y(x_0) = y_0\). Common mistakes include forgetting the chain rule, not substituting correctly, and comparing before fully simplifying. Always show your work clearly—write out the derivative explicitly, show the substitution step, and simplify carefully. Use proper notation and state your conclusion clearly. Practice with different types: exponential functions, polynomials, trigonometric functions, products, and second-order DEs. Verification appears on virtually every AP® exam, so master this fundamental skill! 🎯✨