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Chapter 12: Q76SE (page 539)

In Problems 27鈥�30 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\(\frac{{dy}}{{dx}} - 2xy = {e^{{x_z}}}; y = {e^{{x^2}}}\int_0^x {{e^{t - {t^2}}}} dt\)

Short Answer

Expert verified

The indicated function is a solution of the differential function.

Step by step solution

01

Simplify the given differential equation.

Let the given differential equation be\({\rm{y = }}{{\rm{e}}^{{{\rm{x}}^{\rm{2}}}}}\int_{\rm{0}}^{\rm{x}} {{{\rm{e}}^{{\rm{t - }}{{\rm{t}}^{\rm{2}}}}}} {\rm{dt}}\).

Multiply each side of the equation by\({{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}\).

\(\begin{array}{l}{\rm{y}}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}{\rm{ = }}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}{{\rm{e}}^{{{\rm{x}}^{\rm{2}}}}}\int_{\rm{0}}^{\rm{x}} {{{\rm{e}}^{{\rm{t - }}{{\rm{t}}^{\rm{2}}}}}} {\rm{\;dt}}\\{\rm{y}}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}{\rm{ = }}\int_{\rm{0}}^{\rm{x}} {{{\rm{e}}^{{\rm{t - }}{{\rm{t}}^{\rm{2}}}}}} {\rm{\;dt}}\end{array}\)

02

Determine the solution of the indicated function.

Take differential on both sides of the equation.

\(\begin{array}{c}\frac{{\rm{d}}}{{{\rm{\;dx}}}}\left( {{\rm{y}}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}} \right){\rm{ = }}\frac{{\rm{d}}}{{{\rm{dx}}}}\int_{\rm{0}}^{\rm{x}} {{{\rm{e}}^{{\rm{t - }}{{\rm{t}}^{\rm{2}}}}}} {\rm{\;dt}}\\\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\left( {{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}} \right){\rm{ - 2xy}}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}{\rm{ = }}{{\rm{e}}^{{\rm{x - }}{{\rm{x}}^{\rm{2}}}}}\end{array}\)

Multiply\({{\rm{e}}^{{{\rm{x}}^{\rm{2}}}}}\)on both sides of the equation.

\(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ - 2xy = }}{{\rm{e}}^{\rm{x}}}\)

Hence, the indicated function is a solution of the differential function.

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Most popular questions from this chapter

In Problems \(1\) and \(2\) Fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol \({c_1}\) and has the form \(dy/dx = f(x,y)\). The symbol \({c_1}\) represents a constant.

\(\frac{d}{{dx}}{c_1}{e^{10x}} = \_\_\_\_\_\)

In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\((y - x)y' = y - x + 8;y = x + 4\sqrt {x + 2} \)

In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\(y' = 2x{y^2};y = 1/(4 - {x^2})\)

(a) Verify that the one-parameter family \({y^2} - 2y = {x^2} - x + c\) is an implicit solution of the differential equation \((2y - 2)y' = 2x - 1\).

(b) Find a member of the one-parameter family in part (a) that satisfies the initial condition \(y(0) = 1\).

(c) Use your result in part (b) to and an explicit function \(y = \phi (x)\) that satisfies \(y(0) = 1\). Give the domain of the function \(\phi \). Is \(y = \phi (x)\) a solution of the initial-value problem? If so, give its interval \(I\) of definition; if not, explain.

In Problems \(1 - 8\) state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with \((6)\).

\((sin\theta )y''' - (cos\theta )y' = 2\)
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