Verifying Solutions to Differential…

Part 1 · Hook

Why this matters

A friend claims that

y = 3e^{2x}

fits the rule "the slope is always twice the height". You do not need to take her word for it, and you do not need any new method. Differentiate, substitute, and see whether the two sides match. That check is the whole topic, and it is how you will test every answer in this unit.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is $\frac{d}{dx}\left(e^{3x}\right)$ ?

$3e^{3x}$
$e^{3x}$
$3xe^{3x - 1}$
$\frac{e^{3x}}{3}$

Show the answer

Chain rule: the derivative of the inside, 3, multiplies $e^{3x}$ .

Correct: $3e^{3x}$ :
$e^{3x}$ :
$3xe^{3x - 1}$ :
$\frac{e^{3x}}{3}$ :

2. If $y = \sin(2x)$ , what is $y''$ ?

$-4\sin(2x)$
$4\sin(2x)$
$-2\sin(2x)$
$2\cos(2x)$

Show the answer

$y' = 2\cos(2x)$ , then $y'' = -4\sin(2x)$ .

Correct: $-4\sin(2x)$ :
$4\sin(2x)$ :
$-2\sin(2x)$ :
$2\cos(2x)$ :

Part 4 · See it

See it first

Five curves y = x + 1 + Ce^x for C = −1, −0.5, 0, 0.5 and 1 on −3 ≤ x ≤ 2. The C = 0 curve is the line y = x + 1, drawn in crimson. At x = 0.5 each curve carries a short black segment whose slope is y − 0.5, the value of y − x there, showing that every curve satisfies dy/dx = y − x. — Every curve $y = x + 1 + Ce^x$ satisfies $\frac{dy}{dx} = y - x$ . At $x = 0.5$ each short segment has slope $y - x$ , and each lies along its curve. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

A function is proposed for a differential equationcompute every derivative the equation uses
Substitute the function and its derivatives into both sidessimplify each side separately, never moving terms across the $=$ sign
The two sides agree for every $x$ in an intervalthe function is a solution on that interval
They disagree for even one value of x in the intervalthe function is not a solution there
An added constant has derivative 0one differential equation usually has infinitely many solutions

Part 6 · Key ideas

Key ideas

A solution to a differential equation is a function, not a number. It makes the equation true for every $x$ in an interval.
To verify a solution: differentiate, substitute into both sides, simplify each side, compare.
Checking a single point is not enough: two different expressions can agree at one $x$ .
One equation can have a whole family of solutions, such as $y = Ce^{3x}$ for $\frac{dy}{dx} = 3y$ .
A solution may live on an interval only: $y = \frac{1}{x}$ solves $\frac{dy}{dx} = -y^2$ on $x > 0$ and on $x < 0$ , not across $x = 0$ .

Part 7 · Misconception

A common mistake

The wrong idea: $y = x + 1 + 2e^x$ solves $\frac{dy}{dx} = y - x$ because at $x = 0$ both sides equal 3.

What actually happens: One matching point proves nothing. Compare the sides as functions: $\frac{dy}{dx} = 1 + 2e^x$ and $y - x = 1 + 2e^x$ for every $x$ . That is the verification.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Four candidate functions

Four functions of x, to test against differential equations
Name	$y_1$	$y_2$	$y_3$	$y_4$
Formula	$e^{3x}$	$5e^{3x}$	$e^{3x} + 2$	$3e^{x}$

1. Which functions in the table satisfy $\frac{dy}{dx} = 3y$ ? Select all that apply.

$y_1$
$y_2$
$y_3$
$y_4$

Show the answer

Differentiate each function and compare with $3y$ ; only $y_1$ and $y_2$ match for every $x$ .

Correct: $y_1$ : $\frac{dy}{dx} = 3e^{3x}$ and $3y = 3e^{3x}$ : equal.
Correct: $y_2$ : $\frac{dy}{dx} = 15e^{3x}$ and $3y = 15e^{3x}$ : equal.
$y_3$ : $\frac{dy}{dx} = 3e^{3x}$ but $3y = 3e^{3x} + 6$ : not equal.
$y_4$ : $\frac{dy}{dx} = 3e^{x}$ but $3y = 9e^{x}$ : not equal.

2. Why does $y_3$ fail to satisfy $\frac{dy}{dx} = 3y$ ?

Its derivative is $3e^{3x} + 2$ , which is not $3y_3$ .
At $x = 0$ , $y_3 = 3$ , and 3 is not a rate.
It has the same derivative as $y_1$ , so it is the same function.
Its derivative is $3e^{3x}$ , but $3y_3 = 3e^{3x} + 6$ .

Show the answer

$\frac{dy_3}{dx} = 3e^{3x}$ while $3y_3 = 3e^{3x} + 6$ ; the sides differ by 6 for every $x$ .

Its derivative is $3e^{3x} + 2$ , which is not $3y_3$ .: The derivative of the constant 2 is 0, not 2.
At $x = 0$ , $y_3 = 3$ , and 3 is not a rate.: Values and rates are both numbers; the test is whether the two sides agree.
It has the same derivative as $y_1$ , so it is the same function.: Equal derivatives mean the functions differ by a constant, here 2.
Correct: Its derivative is $3e^{3x}$ , but $3y_3 = 3e^{3x} + 6$ .: The constant 2 drops out of the derivative but is tripled on the right side.

Graph

Three curves $y = x + 1 + Ce^x$

3. Which work verifies that every curve $y = x + 1 + Ce^x$ satisfies $\frac{dy}{dx} = y - x$ ?

At $x = 0$ , $\frac{dy}{dx} = 1 + C$ and $y - x = 1 + C$ .
$\frac{dy}{dx}$ $= 1$ $+ Ce^x$ $= x$ $+ 1$ $+ Ce^x$ $= y$ .
$\frac{dy}{dx} = Ce^x$ and $y - x = 1 + Ce^x$ , so they differ by 1.
$\frac{dy}{dx} = 1 + Ce^x$ and $y - x = 1 + Ce^x$ , equal for every $x$ .

Show the answer

Differentiate: $\frac{dy}{dx} = 1 + Ce^x$ . Substitute: $y - x$ $= (x + 1 + Ce^x) - x$ $= 1$ $+ Ce^x$ . The sides agree for every $x$ .

At $x = 0$ , $\frac{dy}{dx} = 1 + C$ and $y - x = 1 + C$ .: Agreement at one point does not show agreement for every x.
$\frac{dy}{dx}$ $= 1$ $+ Ce^x$ $= x$ $+ 1$ $+ Ce^x$ $= y$ .: This chain writes $=$ between unequal things: $1 + Ce^x \ne x + 1 + Ce^x$ .
$\frac{dy}{dx} = Ce^x$ and $y - x = 1 + Ce^x$ , so they differ by 1.: The derivative of $x$ is 1, which was dropped.
Correct: $\frac{dy}{dx} = 1 + Ce^x$ and $y - x = 1 + Ce^x$ , equal for every $x$ .: Both sides are computed separately and agree as functions.

4. The curve through $(0, 3)$ is $y = x + 1 + 2e^x$ . Find $\frac{dy}{dx}$ at $(0, 3)$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

From the differential equation, $\frac{dy}{dx}$ $= y - x$ $= 3 - 0$ $= 3$ ; from the formula, $1 + 2e^0 = 3$ . They agree.

Answer: 3

5. Is $y = x^2$ a solution to $x\frac{dy}{dx} = 2y$ for $x > 0$ ?

No: $\frac{dy}{dx} = 2x$ , which is not $2y$ .
Yes: at $x = 1$ both sides equal 2.
No: an $x$ on the left side is not allowed.
Yes: $x \cdot 2x = 2x^2$ and $2y = 2x^2$ .

Show the answer

Left: $x \cdot 2x = 2x^2$ . Right: $2x^2$ . They agree for every $x > 0$ , so $y = x^2$ is a solution.

No: $\frac{dy}{dx} = 2x$ , which is not $2y$ .: The left side is $x\frac{dy}{dx}$ , not $\frac{dy}{dx}$ .
Yes: at $x = 1$ both sides equal 2.: The conclusion is right, but one point does not verify it.
No: an $x$ on the left side is not allowed.: A differential equation may contain x anywhere.
Correct: Yes: $x \cdot 2x = 2x^2$ and $2y = 2x^2$ .: Both sides simplify to the same function.

6. For which positive constant $k$ does $y = e^{kx}$ satisfy $y'' - y' - 6y = 0$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$y' = ke^{kx}$ , $y'' = k^2e^{kx}$ . Then $e^{kx}(k^2 - k - 6) = 0$ , and $e^{kx} \ne 0$ , so $(k - 3)(k + 2) = 0$ : $k = 3$ .

Answer: 3

Part 9 · Summary

Summary

To verify a solution to a differential equation, differentiate the function, substitute into both sides and show they agree for every

x

in an interval. One equation can have infinitely many solutions, and a solution may hold only on an interval where everything is defined.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

Four candidate functions

Three curves y=x+1+Cexy = x + 1 + Ce^x

Summary

What comes next

Connections

Three curves $y = x + 1 + Ce^x$