Maths · Ordinary differential equations, their order and degree
The family of curves represented by and the family represented by
The family of curves represented by \( \frac{d y_{1}}{d x}=\frac{x^{2}+x+1}{y^{2}+y+1} \) and the family represented by \( \frac{\boldsymbol{d} \boldsymbol{y}_{2}}{\boldsymbol{d} \boldsymbol{x}}+\frac{\boldsymbol{y}^{2}+\boldsymbol{y}+\mathbf{1}}{\boldsymbol{x}^{2}+\boldsymbol{x}+\mathbf{1}}=\mathbf{0} \)
- A. touch each other
- B. orthogonal to each other
- C. identical
- D. intersect at an angle of \( \frac{\pi}{4} \)
Step-by-step solution
The slopes of the two families are m1 = (x^2+x+1)/(y^2+y+1) and m2 = -(y^2+y+1)/(x^2+x+1). Their product m1 * m2 = -1, which indicates that the curves are orthogonal (perpendicular) at points of intersection.
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