Chapter 1 - First-Order Differential Equations - 1.8 Change of Variables - Problems - Page 79: 31

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Given: $(x-c)^2+(y-c)^2=2c^2$ Differentiate: $2(x-c)+2(y-c)\frac{dy}{dx}=0\\ \rightarrow \frac{dy}{dx}=\frac{c-x}{y-c}\\ \rightarrow \frac{dy}{dx}=-\frac{y-c}{c-x}\\ \rightarrow \frac{dy}{y-c}=-\frac{dx}{c-x}$ Integrate: $\int \frac{dy}{y-c}=-\frac{dx}{c-x}\\ \rightarrow \ln (y-c)=\ln(x-c)+\ln k\\ \rightarrow y=(x-c)c_1$ The family of orthogonal trajectories can be $y=(x-c)c_1$