91Ó°ÊÓ

We have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. $$ \frac{d y}{d x}=x+y $$

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\frac{1}{x^{2}} ; y(1)=5\)

A more detailed version of Theorem 1 says that, if the function \(f(x, y)\) is continuous near the point \((a, b)\), then at least one solution of the differential equation \(y^{\prime}=f(x, y)\) exists on some open interval I containing the point \(x=a\) and, moreover, that if in addition the partial derivative \(\partial f / \partial y\) is continuous near \((a, b)\), then this solution is unique on some (perhaps smaller) interval \(J .\) In Problems 11 through 20, determine whether \(e x\) istence of at least one solution of the given initial value problem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. $$ \frac{d y}{d x}=2 x^{2} y^{2} ; \quad y(1)=-1 $$

Substitute \(y=e^{r x}\) into the given differential equation to determine all values of the constant \(r\) for which \(y=e^{r x}\) is a solution of the equation. $$ 4 y^{\prime \prime}=y $$

First verify that \(y(x)\) satisfies the given differential equation. Then determine a value of the constant \(C\) so that \(y(x)\) satisfies the given initial condition. Use a computer or graphing calculator ( if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. $$ y^{\prime}+y=0 ; y(x)=C e^{-x}, y(0)=2 $$

A function \(y=g(x)\) is described by some geometric property of its graph. Write a differential equation of the form \(d y / d x=f(x, y)\) having the function \(g\) as its solution (or as one of its solutions). The slope of the graph of \(g\) at the point \((x, y)\) is the sum of \(x\) and \(y\).

Write-in the manner of Eqs. (3) through (6) of this section-a differential equation that is a mathematical model of the situation described. The time rate of change of the velocity \(v\) of a coasting motorboat is proportional to the square of \(v\).

Determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. $$ x y^{\prime}+y=3 x^{2} $$

Suppose that a mineral body formed in an ancient cataclysm-perhaps the formation of the earth itselforiginally contained the uranium isotope \({ }^{238} \mathrm{U}\) (which has a half-life of \(4.51 \times 10^{9}\) years) but no lead, the end product of the radioactive decay of \({ }^{238} \mathrm{U}\). If today the ratio of \({ }^{238} \mathrm{U}\) atoms to lead atoms in the mineral body is \(0.9\), when did the cataclysm occur?

Determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your hypothesis. $$ y^{\prime \prime}+y=0 $$