Class 12

Math

Calculus

Differential Equations

Show that the differential equation $(x−y)dxdy =x+2y$is homogeneous and solve it.

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Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.$y_{2}=a(b_{2}−x_{2})$

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.$y=ae_{3x}+be_{−2x}$

Solve the differential equation $(tan_{−1}y−x)dy=(1+y_{2})dx$.

Prove that $x_{2}−y_{2}=c(x_{2}+y_{2})_{2}$is the general solution of differential equation $(x_{3}−2xy_{2})dx=(y_{3}−3x_{2}y)dy$, where c is a parameter.

Find the particular solution of the differential equation $(1+e_{2x})dy+(1+y_{2})e_{x}dx=0$, given that $y=1whenx=0$.

Find the order and degree, if defined, of each of the following differential equations:(i) $dxdy −cosx=0$ (ii) $xydx_{2}d_{2}y +x(dxdy )_{2}−ydxdy =0$ (iii) $y_{primeprimeprime}+y_{2}+e_{y_{p}rime}=0$

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Find the general solution of the differential equations $e_{x}tanydx+(1−e_{x})sec_{2}ydy=0$