Solve this differential equation---Form the differential equation representing the family of curves y = e^(2x)(a + bx), where 'a' and 'b' are arbitrary constants. (Delhi 2019)
视频信息
答案文本
视频字幕
We need to form a differential equation from the family of curves y equals e to the power 2x times the quantity a plus bx, where a and b are arbitrary constants. Our goal is to eliminate these constants to get a differential equation that represents the entire family.
Step 1: We differentiate the given equation with respect to x using the product rule. The derivative of e to the 2x times the quantity a plus bx equals 2 e to the 2x times a plus bx, plus e to the 2x times b. Simplifying, we get y prime equals 2 e to the 2x times a plus bx, plus b e to the 2x. Since the original equation tells us that y equals e to the 2x times a plus bx, we can substitute to get y prime equals 2y plus b e to the 2x.
Step 2: Now we differentiate the first derivative to get the second derivative. From y prime equals 2y plus b e to the 2x, we get y double prime equals 2y prime plus 2b e to the 2x. From the first derivative equation, we can isolate b e to the 2x as y prime minus 2y. Substituting this back, we get y double prime equals 2y prime plus 2 times the quantity y prime minus 2y. Simplifying gives us y double prime equals 4y prime minus 4y, which rearranges to our final differential equation: y double prime minus 4y prime plus 4y equals zero.
Let's verify our differential equation by substituting the original family back into it. Starting with y equals e to the 2x times a plus bx, we have y prime equals 2 e to the 2x times a plus bx plus b e to the 2x, and y double prime equals 4 e to the 2x times a plus bx plus 4b e to the 2x. When we substitute into y double prime minus 4y prime plus 4y, we get 4 e to the 2x times a plus bx plus 4b e to the 2x, minus 4 times the quantity 2 e to the 2x times a plus bx plus b e to the 2x, plus 4 times e to the 2x times a plus bx. All terms cancel out, giving us zero, confirming our differential equation is correct.
In summary, we have successfully formed the differential equation representing the family of curves y equals e to the 2x times the quantity a plus bx. By differentiating twice and eliminating the arbitrary constants a and b, we obtained the second-order linear homogeneous differential equation: y double prime minus 4y prime plus 4y equals zero. This differential equation represents the entire family of curves, and any solution to this equation will be of the form y equals e to the 2x times a plus bx for some constants a and b.