Sometimes a change of variable can be used to convert a differential equation y′=f(t,y) into a separable equation. One common change of variable technique is as follows. Consider a differential equation of the form y′=f(αt+βy+γ), where α,β, and γ are constants. Use the change of variable z=αt+βy+γ to rewrite the differential equation as a separable equation of the form z′=g(z). Solve the initial value problem y′=(t+y)2−1, y(3)=4.

Answer:[tex]y=\frac{-7t^2+22t-7}{7t-22}[/tex]Step-by-step explanation:We are given that Initial value problem [tex]y'=(t+y)^2-1[/tex], y(3)=4Substitute the value [tex]z=t+y[/tex]When t=3 and y=4 then z=3+4=7[tex]y'=z^2-1[/tex]Differentiate z w.r.t tThen, we get [tex]\frac{dz}{dt}=1+y'[/tex][tex]z'=1+z^2-1=z^2[/tex][tex]z^{-2}dz=dt[/tex]Integrate on both sides [tex]-\frac{1}{z}dz=t+C[/tex][tex]z=-\frac{1}{t+C}[/tex]Substitute t=3 and z=7Then, we get[tex]7=-\frac{1}{3+C}[/tex][tex]21+7C=-1[/tex][tex]7C=-1-21=-22[/tex][tex]C=-\frac{22}{7}[/tex]Substitute the value of C then we get [tex]z=-\frac{1}{t-\frac{22}{7}}[/tex][tex]z=\frac{-7}{7t-22}[/tex][tex]y=z-t[/tex][tex]y=\frac{-7}{7t-22}-t[/tex][tex]y=\frac{-7-7t^2+22t}{7t-22}[/tex][tex]y=\frac{-7t^2+22t-7}{7t-22}[/tex]