Participation Assignment $-$ Section $2\mathrm{.}5-$ Substitution

Name:

The overall goal is to solve the IVP of $\frac{dP}{dt}=co{s}^2(P+t)-1$ where $P(0)=0,$ by working through the questions below.

What is the appropriate substitution to take to simplify the differential equation? Use $u$ for the substitution variable.

Based on the substitution from question $1,$ what will be the substitution used for the $\frac{dP}{dt}$ in the differential equation?

After making those substitutions, answer the following two sub$-$questions...

a$.$ Under the substitution, what does the differential equation simplify to$?$

b$.$ What solving strategy will be easiest to use from here?

Participation Assignment $-$ Section $2\mathrm{.}5-$ Substitution

Name:

The overall goal is to solve the IVP of $\frac{dP}{dt}=co{s}^2(P+t)-1$ where $P(0)=0,$ by working through the questions below.

What is the appropriate substitution to take to simplify the differential equation? Use $u$ for the substitution variable.

Based on the substitution from question $1,$ what will be the substitution used for the $\frac{dP}{dt}$ in the differential equation?

After making those substitutions, answer the following two sub$-$questions...

a$.$ Under the substitution, what does the differential equation simplify to$?$

b$.$ What solving strategy will be easiest to use from here?

Solve the IVP for $P$ as a function of $t($leave you answer in implicit form$).$

What is the interval of definition?