Solution of Ex:4.1 Differential Equations with Boundary-Value Problems (7th Edition Book) consist of initial and boundary value problems. In this post, we will provide solved questions of Ex:4.1 in hard form given below.
Solution of Ex:4.1 Differential Equations with Boundary-Value Problems (7th Edition Book)
Exercise
![\[y'' + y = 0, y(0) = 0, y(pi/2) = 1\]](https://techuros.com/wp-content/ql-cache/quicklatex.com-1fb0ed233a7743d5d50d080c4d08763e_l3.png "Rendered by techuros.com")
We first find the GS (general Solution) to the (DE) differential equation y” + y = 0.
The general solution is
![\[y = c1 cos(x) + c2 sin(x),\]](https://techuros.com/wp-content/ql-cache/quicklatex.com-ab90d71cafe1e837ae7e41acdfea5e1e_l3.png "Rendered by techuros.com")
where c1 and c2 are constants that depend on the boundary conditions.
Boundary conditions (BCs) are y(0) = 0 and y(pi/2) = 1
Substituting x = 0 , we get
![\[0 = c1 cos(0) + c2 sin(0),\]](https://techuros.com/wp-content/ql-cache/quicklatex.com-78185158e16b484e729c34cbb968092b_l3.png "Rendered by techuros.com")
![\[c1 = 0.\]](https://techuros.com/wp-content/ql-cache/quicklatex.com-f18f24f9c8e2a481d05b1c03643d74d1_l3.png "Rendered by techuros.com")
Substituting 
, we get
![\[1 = c2 sin(pi/2),\]](https://techuros.com/wp-content/ql-cache/quicklatex.com-caf3dee5fe1dea7f0ba16bed37cc55d9_l3.png "Rendered by techuros.com")
![\[c2 = 1.\]](https://techuros.com/wp-content/ql-cache/quicklatex.com-570879cd8deece90cf7998fad5795fb7_l3.png "Rendered by techuros.com")
Therefore, the solution to the BVP is 
We can check that this solution satisfies the differential equation and the boundary conditions.
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