{"id":330,"date":"2023-09-20T22:49:08","date_gmt":"2023-09-20T22:49:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/initial-value-problems\/"},"modified":"2024-08-05T13:35:48","modified_gmt":"2024-08-05T13:35:48","slug":"antiderivatives-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/antiderivatives-learn-it-3\/","title":{"raw":"Antiderivatives: Learn It 3","rendered":"Antiderivatives: Learn It 3"},"content":{"raw":"

Initial-Value Problems<\/h2>\r\n

We look at techniques for integrating a large variety of functions involving products, quotients, and compositions later in the text. Here we turn to one common use for antiderivatives that arises often in many applications: solving differential equations.<\/p>\r\n

A differential equation<\/em> is an equation that relates an unknown function and one or more of its derivatives. The equation [latex]\\frac{dy}{dx}=f(x)[\/latex] is a simple example of a differential equation.<\/p>\r\n

Solving this equation means finding a function [latex]y[\/latex] with a derivative [latex]f[\/latex]. Therefore, the solutions of [latex]\\frac{dy}{dx}[\/latex] are the antiderivatives of [latex]f[\/latex]. If [latex]F[\/latex] is one antiderivative of [latex]f[\/latex], every function of the form [latex]y=F(x)+C[\/latex] is a solution of that differential equation.<\/p>\r\n

\r\n

The solutions of<\/p>\r\n

[latex]\\frac{dy}{dx}=6x^2[\/latex]<\/div>\r\n

are given by<\/p>\r\n

[latex]y=\\displaystyle\\int 6x^2 dx=2x^3+C[\/latex]<\/div>\r\n<\/section>\r\n

Sometimes we are interested in determining whether a particular solution curve passes through a certain point [latex](x_0,y_0)[\/latex]\u2014that is, [latex]y(x_0)=y_0[\/latex]. The problem of finding a function [latex]y[\/latex] that satisfies a differential equation [latex]\\frac{dy}{dx}=f(x)[\/latex] with the additional condition [latex]y(x_0)=y_0[\/latex] is an example of an initial-value problem<\/strong>. The condition [latex]y(x_0)=y_0[\/latex] is known as an initial condition<\/em>.<\/p>\r\n

\r\n

Looking for a function [latex]y[\/latex] that satisfies the differential equation<\/p>\r\n

[latex]\\frac{dy}{dx}=6x^2[\/latex]<\/div>\r\n

 <\/p>\r\n

and the initial condition<\/p>\r\n

[latex]y(1)=5[\/latex]<\/div>\r\n\r\nis an example of an initial-value problem.\r\n\r\n

Since the solutions of the differential equation are [latex]y=2x^3+C[\/latex], to find a function [latex]y[\/latex] that also satisfies the initial condition, we need to find [latex]C[\/latex] such that [latex]y(1)=2(1)^3+C=5[\/latex]. From this equation, we see that [latex]C=3[\/latex], and we conclude that [latex]y=2x^3+3[\/latex] is the solution of this initial-value problem as shown in the following graph.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"641\"]\"The Figure 2. Some of the solution curves of the differential equation [latex]\\frac{dy}{dx}=6x^2[\/latex] are displayed. The function [latex]y=2x^3+3[\/latex] satisfies the differential equation and the initial condition [latex]y(1)=5[\/latex].[\/caption]\r\n<\/section>\r\n

\r\n

Solve the initial-value problem<\/p>\r\n

[latex]\\frac{dy}{dx}= \\sin x, \\,\\,\\, y(0)=5[\/latex]<\/div>\r\n\r\n[reveal-answer q=\"fs-id1165043430929\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"fs-id1165043430929\"]\r\n\r\n

First we need to solve the differential equation. If [latex]\\frac{dy}{dx}= \\sin x,[\/latex] then<\/p>\r\n

[latex]y=\\displaystyle\\int \\sin (x) dx=\u2212 \\cos x+C[\/latex]<\/div>\r\n

Next we need to look for a solution [latex]y[\/latex] that satisfies the initial condition. The initial condition [latex]y(0)=5[\/latex] means we need a constant [latex]C[\/latex] such that [latex]\u2212 \\cos x+C=5[\/latex]. Therefore,<\/p>\r\n

[latex]C=5+ \\cos (0)=6[\/latex]<\/div>\r\n

The solution of the initial-value problem is [latex]y=\u2212 \\cos x+6[\/latex].<\/p>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

\r\n

Solve the initial value problem [latex]\\frac{dy}{dx}=3x^{-2}, \\,\\,\\, y(1)=2[\/latex].<\/p>\r\n

[reveal-answer q=\"7073552\"]Hint[\/reveal-answer]
\r\n[hidden-answer a=\"7073552\"]<\/p>\r\n

Find all antiderivatives of [latex]f(x)=3x^{-2}[\/latex].<\/p>\r\n

[\/hidden-answer]<\/p>\r\n

[reveal-answer q=\"fs-id1165043327482\"]Show Solution[\/reveal-answer]<\/p>\r\n

[hidden-answer a=\"fs-id1165043327482\"]<\/p>\r\n

[latex]y=-\\frac{3}{x}+5[\/latex]<\/p>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

\r\n

Watch the following video to see the worked solution to the two previous examples.<\/p>\r\n