Question

Find the particular solution to the differential equation
6(du)/(dt)=u^2
subject to the initial condition u(0)=8

Answer 1

This is separable.$$6\frac{du}{dt}=u^2\\6\,du=u^2\, dt\\\frac1{u^2}du=\frac16dt\\\int\frac1{u^2}du=\int\frac16dt\\-\frac1u=\frac16t+C\\u=-\frac1{\frac16t+C}=-\frac6{t+C}\text{ by multiplying by }\frac66$$

Answer 2

Then we just use our initial condition \(u(0)=8\) to figure out \(C\):$$u(0)=8\\-\frac6{0+C}=8\\-\frac6C=8\\8C=-6\\C=-\frac34$$

Answer 3

Ok

Answer 4

Thank you!