# Solve for y(x) using the Fundamental Theorem of Calculus

## Homework Statement

Solve the integral equation for y(x):
y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt
(integral from 0 to x)

See attached image for the equation in a nicer format.

## Homework Equations

Fundamental Theorem of Calculus

## The Attempt at a Solution

dy/dx = y(x)^2 / (1 + x^2)
∫ dy/y^2 = ∫ dx/(1 + x^2)
-1/y = arctan(x) + C
y = -1/arctan(x) + C

y(0) = 1 = -1/arctan(0) + C
C = 1 + 1/arctan(0)
Since arctan(0) = 0, the fraction is indeterminate.

In lectures we only did one practice example, so if anyone could give me a hand and point out where I am going wrong, it'll be greatly appreciated.

#### Attachments

• Calculus Equation.png
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Dick
Homework Helper

## Homework Statement

Solve the integral equation for y(x):
y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt
(integral from 0 to x)

See attached image for the equation in a nicer format.

## Homework Equations

Fundamental Theorem of Calculus

## The Attempt at a Solution

dy/dx = y(x)^2 / (1 + x^2)
∫ dy/y^2 = ∫ dx/(1 + x^2)
-1/y = arctan(x) + C
y = -1/arctan(x) + C

y(0) = 1 = -1/arctan(0) + C
C = 1 + 1/arctan(0)
Since arctan(0) = 0, the fraction is indeterminate.

In lectures we only did one practice example, so if anyone could give me a hand and point out where I am going wrong, it'll be greatly appreciated.

Use parentheses! -1/arctan(x)+C is not the same thing as -1/(arctan(x)+C)! One is indeterminant, one is -1/C. Which one do you want?

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Solve the integral equation for y(x):
y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt
(integral from 0 to x)

See attached image for the equation in a nicer format.

## Homework Equations

Fundamental Theorem of Calculus

## The Attempt at a Solution

dy/dx = y(x)^2 / (1 + x^2)
∫ dy/y^2 = ∫ dx/(1 + x^2)
-1/y = arctan(x) + C
y = -1/(arctan(x) + C)

y(0) = 1 = -1/arctan(0) + C
C = 1 + 1/arctan(0)
Since arctan(0) = 0, the fraction is indeterminate.
This is a good example of why you need to not be sloppy about using parentheses.

: Dick wins again.

Use parentheses! -1/arctan(x)+C is not the same thing as -1/(arctan(x)+C)! One is indeterminant, one is -1/C. Which one do you want?
I definitely want -1/C. Thank you for helping me figure it out. I knew it was something small but I couldn't see it.

This is a good example of why you need to not be sloppy about using parentheses.

: Dick wins again.
It wasn't that I was sloppy. I commonly leave the C on the outer edge of my equations because C cannot loss its generality. But it definitely reminded me to use the generality to my advantage. Thank you to Dick and yourself for helping students with their problems.

Dick