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Worked solution

Integrate, apply an initial value and sketch

Find the general solution of the differential equation dydx=10x−3x2\dfrac{dy}{dx}=10x-3x^2, expressing yy in terms of xx. Given that y=100y=100 when x=0x=0, sketch this curve.

CJC · Discussion · Question 1

Answer

y=5x2−x3+Ay=5x^2-x^3+A; with the condition, y=5x2−x3+100y=5x^2-x^3+100, sketched below.

Integrate the derivative

y=5x2−x3+A,y=5x^2-x^3+A,

where A is an arbitrary constant.

How did this part go?

Use the initial value

At x=0x=0, 100=5(0)2−0+A100=5(0)^2-0+A, so A=100A=100. The particular solution is

y=5x2−x3+100.y=5x^2-x^3+100.

How did this part go?

Sketch the particular solution

The source calculator enters Y1=5X2−X3+100Y_1=5X^2-X^3+100, with window −10≤X≤10-10\le X\le10, −20≤Y≤200-20\le Y\le200 and unit axis scales. The curve passes through (0,100)(0,100), has a minimum there and a maximum at (10/3,3200/27)(10/3,3200/27). Its x-intercept is approximately 7.03. These labels are reproduced in the answer sketch; derivative x(10−3x)x(10-3x) verifies both turning points.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Exact polynomial y=5x²-x³+100, local minimum (0,100), local maximum (10/3,3200/27), positive x-intercept approximately 7.03, rising to the left and falling to the right. Finite display window; curve continues outside it.
Exact polynomial y=5x²-x³+100, local minimum (0,100), local maximum (10/3,3200/27), positive x-intercept approximately 7.03, rising to the left and falling to the right. Finite display window; curve continues outside it.
Exact polynomial y=5x²-x³+100, local minimum (0,100), local maximum (10/3,3200/27), positive x-intercept approximately 7.03, rising to the left and falling to the right. Finite display window; curve continues outside it.

Common mistakes

The requested general solution still needs its arbitrary constant.

Solution notes

Both general and particular solutions, calculator entry/window and labelled source sketch retained. The curve is independently plotted from the exact polynomial.

Read the hint