Differential Equation and Applications (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question
Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Differential Equation and Applications (p-1)3 Marks
Question
Solve $\left( x +2 y ^3\right) \frac{d y}{d x}= y$
✓
Answer
$ \begin{gathered} \left( x +2 y ^3\right) \frac{d y}{d x}= y \\ \therefore x +2 y ^3= y \frac{d x}{d y} \\ \therefore \frac{x}{y}+2 y^2=\frac{d x}{d y} \\ \therefore \frac{d x}{d y}-\frac{x}{y}=2 y^2 \end{gathered} $ This is a linear differential equation of the form $\frac{d x}{d y}+P x=Q$, where $P=-\frac{1}{y}, Q=2 y^2$ $ \therefore \text { I.F. }=e^{\int P d y}=e^{\int-\frac{1}{y} d y}=e^{-\int \frac{1}{y} d y} $ $ =e^{-\log y}=e^{\log \left(\frac{1}{y}\right)}=\frac{1}{y} $ $\therefore$ the solution of (1) is given by $ \begin{aligned} & x \cdot(\text { I.F. })=\int Q \cdot(\text { I.F. }) d y+c \\ & \therefore \frac{x}{y}=\int 2 y^2 \cdot \frac{1}{y} d y+c=2 \int y d y+c \\ & \quad=2\left(\frac{y^2}{2}\right)+c \\ & \therefore x=y\left(y^2+c\right) \end{aligned} $ This is the general solution.
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