Q.Find the equation of a curve passing through the point $(0, 2)$ given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

Concept: Initial Value Problem — we translate the geometric condition into a differential equation and then use the given point to find the particular solution.

Let the curve be $y = f(x)$. At any point $(x, y)$, the slope of the tangent is $\frac{dy}{dx}$. The condition says:

Since the curve passes through $(0,2)$, near this point $x+y$ is positive, so we take the positive case:

This is a linear differential equation: $\frac{dy}{dx} - y = x - 5$.

Integrating factor: $e^{\int -1\,dx} = e^{-x}$. Multiply:

Integrate both sides: …

At any point $(x, y)$ the slope of the tangent is $\dfrac{dy}{dx}$. The sum of the coordinates, $x + y$, exceeds this slope by $5$:

Write it in linear form $\dfrac{dy}{dx} + P(x)\,y = Q(x)$:

Integrating factor:

Multiply through; the left side becomes an exact derivative:

$\frac{d}{dx}\big(y\,e^{-x}\big) = (x - 5)\,e^{-x}.$ …

Method: Modelling a worded condition, handling the absolute value, then the point

Translate the sentence into a differential equation, resolve any $\left|\dfrac{dy}{dx}\right|$ using the given point, solve, and apply the point.

Steps

Step 1: Turn the words into an equation.

"Sum of coordinates exceeds the slope by $5$" becomes $x+y=\dfrac{dy}{dx}+5$, i.e. $\dfrac{dy}{dx}=x+y-5$.

Step 2: Fix the sign of an absolute value from the point. …

Common Mistakes

Mistake 1: Misordering the "exceeds by 5" relation.

Why it's wrong: "sum of coordinates exceeds the slope by $5$" means $x+y=\dfrac{dy}{dx}+5$, i.e. $\dfrac{dy}{dx}=x+y-5$; reversing it gives the wrong sign. Correct approach: the larger quantity ($x+y$) equals the smaller plus $5$.

Mistake 2: Mishandling the absolute value of the slope.

Why it's wrong: near $(0,2)$ the slope's sign must be chosen consistently before dropping $|\cdot|$. Correct approach: pick the sign that fits the given point, then solve. …