Direction fields and Euler\'s Method Separable equations Consider the differenti

Question

Direction fields and Euler's Method

Separable equations

Consider the differential equation dy/dx = 2y At the point (-1.5, 1), the direction field has a slope of 2 Preview 2 = 2. At the point (1, -1)= the direction field has a slope of -2 Preview -2 = -2. Along the x-axis the slope of the direction field is equal to which number or expression? 0 preview syntax error Along the y-axis the slope of the direction field is equal to which number or expression? Use your answers above to help choose the correct direction field for the differential equation. Consider the differential equation dy/dx = x11/y2 Re-write the equation in terms of differentials: Now integrate each side of the equation: Solve the equation for y , given that that y(0) = 4

Explanation / Answer

(A) y2dy=x11dx

(B) y3/3+C1=x12/12+C2

(C) y3/3=x12/12+(C2-C1)

or, y3=3{x12/12+(C2-C1)}

or, y3=x12/4+3(C2-C1)

or, y={x12/4+3(C2-C1)}(1/3)

given,y(0)=4

As we put x=0 in the equation obtained we get,

y={0+3(C2-C1)}(1/3)=4

or, 3(C2-C1)=43=64

hence,

we can write in general,by substituting for 3(C2-C1)as,

y={x12/4 + 64}(1/3)

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